We show that a reduct of the Zariski structure of an algebraic curve which is not locally modular interprets a field, answering a question of Zilber's.
reducts.tex
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\begin{center}
  {\LARGE\bf Incidence systems on Cartesian powers of algebraic
    curves}\\[2ex] 
\end{center}

\begin{center}
  {\normalsize Assaf Hasson and Dmitry Sustretov \footnote{Has been
      supported by a fellowship of the Center for Advanced Studies in
      Mathematics at Ben-Gurion University of the Negev during the
      work on the article. The research leading to these results has
      received funding from the European Research Council under the
      European Union’s Seventh Framework Programme (FP7/2007-
      2013)/ERC Grant Agreement No.~291111 and No.~615722
      (MOTMELSUM).}

}
\end{center}
\begin{center}
\begin{minipage}[t]{0.9\linewidth}
\begin{abstract}
  We show that a reduct of the Zariski structure of an algebraic curve
  which is not locally modular interprets a field, answering a
  question of Zilber's.
\end{abstract}
\end{minipage}
\newpage
{ \small
\tableofcontents }

\end{center}


\section{Introduction}
In \cite[\S 2]{ArtinGeometry} Artin describes the basic problem of
classifying abstract plane geometries (viewed as incidence systems of
points and lines) as follows ``Given a plane geometry ... assume that
certain axioms of geometric nature are true ... is it possible to find
a field $k$ such that the points of our geometry can be described by
coordinates from $k$ and lines by linear equations?''. Zilber's
trichotomy principle (to be described in more detail in the next
section) can be viewed as an abstraction of the above problem,
replacing the ``axioms of geometric nature'' with a well behaved
theory of dimension (see, e.g., \cite[\S 1]{poizat}).

Conjectured in various forms by Zilber throughout the late 1970s,
essentially every aspect of Zilber's trichotomy, in its full
generality, was refuted by Hrushovski \cite{Hr1}, \cite{Hr2} in the
late 1980s. Due to Hrushovski's zoo of counterexamples the conjecture
has never been reformulated. Yet, Zilber's principle remains a central 
and powerful theme in model theory: it has been proved to hold in many
natural examples such as differentially closed fields of
characteristic $0$, algebraically closed fields with a (generic)
automorphism, o-minimal theories and more (see
\citep{chatzhr,pillay-ziegler,peterzil1998trichotomy,chatzidakis2002model,
  hz}). Many of these special cases of Zilber's trichotomy had 
striking applications in algebra and geometry
(\citep{hrushovski1996mordell,hrushovski2001manin,scanlon2006local}).

The key to the classification of Desarguesian plane geometries (the
fundamental theorem of projective geometry) is the reconstruction of
the underlying field $k$ as the ring of direction preserving
endomorphisms of the group of translations.  The reconstruction of a
field out of abstract geometric data is also the essence of Zilber's
trichotomy and is the engine in many of its applications.  A
relatively recent application of one such result is Zilber's model
theoretic proof \cite{ztorelli} of a significant strengthening of a
theorem of Bogomolov, Korotiaev, and Tschinkel (\cite{jacobian}). The
model theoretic heart of Zilber's proof is Rabinovich' trichotomy
theorem for \emph{reducts} of algebraically closed fields
\cite{rabinovich}. In the concluding paragraph of the introduction to
\cite{ztorelli} Zilber writes: ``It is therefore natural to aim for a
new proof of Rabinovich' theorem, or even a full proof of the
Restrited Trichotomy along the lines of the classification theorem of
Hrushovski and Zilber \cite{hz}, or by other modern methods
[...]. This is a challenge for the model-theoretic community.''

The conjecture referred to in Zilber's text above can be formulated as
follows\footnote{The content of Conjecture \ref{RTC} is explained for
  the non-experts in Section \ref{background}.}: 

\begin{conjecture}\label{RTC}
  Let $\Mm$ be a non-locally modular strongly minimal reduct of the
  full Zariski structure, $\mathfrak M$, on an algebraic curve $M$
  over an algebraically closed field $K$. Then, there exist
  $\Mm$-definable $L,E$ such that $E\subseteq L\times L$ is an
  equivalence relation with finite classes and $L/E$ with the
  $\Mm$-induced structure is a field $K$-definably isomorphic to $K$.
\end{conjecture}

Rabinovich \cite{rabinovich} proved Conjecture \ref{RTC} in the
special case where $M=\A^1$, and her result can be extend by general
principles to any rational curve. In the present paper we prove
Conjecture \ref{RTC}.  Our proof (Theorem~\ref{main-theorem}) is
geometric in nature and does not use advanced model theoretic
machinery. Roughly, it proceeds in four main steps:
\begin{enumerate}
\item Given a non locally modular reduct $\Mm$ of the full Zariski
  structure $\mathfrak M$ on an algebraic curve, fix a $2$-dimensional
  (almost) faithful family $X\subseteq M^2\times T$ of curves in
  $M^2$. Throughout the text we assume that for almost all $t\in T$
  the curve $X_t$ does not have $0$-dimensional irreducible
  components. This assumption considerably simplifies the discussion,
  and is justified in Section~\ref{pure-dimensional}.
\item We introduce the notion of the slope of a curve $C\subseteq M^2$
  at a point $P \in C$, and use it to define when two curves
  $X_t,X_s\in X$ incident to $P$ are tangent at that point and show
  that this geometric notion is $\Mm$-definable up to an
  $\Mm$-definable equivalence relation with finite classes,
  Proposition~\ref{tangency-intersection}.
% I.e., there exist are an $\Mm$-definable set $T_0\subseteq T\times T$ and an $\Mm$-definable equivalence relation $E$ with finite   classes, such that for any $(t,s)\in T_0$ there are $t'Et$ and  $s'Es$ such that $X_{t'}$ is tangent to $X_{s'}$ at $P$, 
\item Using our assumption of the first clause and standard model
  theoretic machinery we reconstruct a 1-dimensional algebraic group
  in $\Mm$, Subsection~\ref{first-group}. To carry out this
  construction we have to assure the existence of enough slopes of
  curves from our family passing through a given point. In
  characteristic $0$ this follows from the uniqueness of solutions of
  ordinary differential equations for formal power series over $K$. In
  characteristic $p>0$ the kernel of derivation is non-trivial so
  extra care is needed in the choice of the family $X$, and we were
  unable to avoid having to work with high-order slopes\footnote{In a
    similar situation Rabinovich, \cite[Section 8, p.93]{rabinovich}
    seems to claim that she can actually recover an additive subgroup
    of $(K^2,+)$, giving rise to a characteristic-independent
    argument.}.
\item This reduces us to proving Conjecture~\ref{RTC} in the context
  where $\Mm$ is a non-locally modular expansion of a 1-dimensional
  algebraic group. We apply (Subection~\ref{second-group}) the tools
  developed in the previous sections to generalise the result of
  \cite{marker-pillay} (addressing the same problem for
  $(\mathbb C,+)$) to the present, fully general, context
  (Theorem~\ref{marker-pillay-plus}).
\end{enumerate}

The general scheme of our proof seems to have much in common with
Rabinovich' original work, though we were unable to understand
significant parts of her argument which are highly technical.  Step
(2) of the above strategy is at the conceptual and technical heart of
the paper, and it relies -- ultimately -- on classical intersection
theory. As Rabinovich's main tool for studying intersections of plane
curves is the classical Bezout theorem, it stands to reason that the
more advanced tools applied in Sub-section \ref{flat-families} are at
the source of the greater generality of the present paper. We also
believe that our more liberal application of algebro-geometric tools
such as non-reduced schemes helped simplify the exposition,
considerably lowering the level of combinatorial complexity.

It seems that the tools developed in the present paper can be extended
to various other contexts. For example, extending the results of
\cite{piotr-serge} to positive characteristic, and any algebraic group
and -- possibly -- even a full proof of the restricted trichotomy
conjecture for structures interpretable in ACVF\footnote{At least
  modulo the problem of showing that the 1-dimensional group
  reconstructed by our methods embeds in an algebraic group.}.

Section \ref{pure-dimensional} seems to have no obvious counterpart in
existing model theoretic literature. We believe that the results of
that section could be applied in any topological context where
definable nowhere locally constant unary functions are open with
finite fibres (see, e.g., \cite[Theorem 5.5]{Whyburn}). It seems that
coupled with analogous results (yet to be proved) for the uniform
definability of the frontier of plane curves a version of Theorem
\ref{cleanup} could be proved for all dimensions, namely, that
reduct-definable reduct-irreducible sets are pure-dimensional. This
could be a key for a shorter proof of the restricted trichotomy
conjecture, using the machinery of Zariski Geometries, possibly in all
dimensions. \\ 
 
\textbf{Acknowledgements.}  The second author thanks Boris Zilber for
his remarks on an early version of the paper, Maxim Mornev for many
helpful comments, and Qiaochu Yuan for suggesting the elegant proof of
Lemma~\ref{ode-unique}. We would also like to thank Moshe Kamensky for
some comments and suggestions.


\section{Model theoretic background}\label{background}

For readers unfamiliar with the model theoretic jargon we give a self
contained explanation of Conjecture \ref{RTC}. In order to keep this
introduction as short as possible, we specialise our definitions to
the setting in which they will be applied. Readers familiar with the
basics of model theory are advised to skip to Subsection
\ref{section:setup}.

\subsection{Zilber's restricted trichotomy conjecture}
\label{intro-RTC}

Throughout this paper we fix an algebraic curve $M$ over a field $k$,
which -- to simplify the discussion -- we assume to be algebraically
closed. We do not assume $M$ to be irreducible or proper. As we will
see, there is no loss of generality in assuming that $M$ is regular,
but we do not a priori assume it. The \emph{full Zariski structure,
  $\mathfrak M$, on $M$} is the collection of all constructible
subsets of Cartesian powers of $M$. Notationally, we will not
distinguish between the curve $M$ as an algebro-geometric object and
its $k$-rational points.

In the context of the present work, this construction should be viewed
as a generalisation of the notion of abstract plane geometry in the
sense of \cite[\S2]{ArtinGeometry}. Incidence systems (within families
of) curves in $\mathfrak M\times \mathfrak M$ will be our main concern
in this paper. As already mentioned if $\mathfrak M$ is $\mathbb A^1$,
then the ground field $k$ can be reconstructed from the incidence
system of affine lines in $\mathbb A^2$. More generally, it follows
from \cite{hz} that $k$ can be reconstructed from the full Zariski
structure $\mathfrak M$.

Zilber's restricted trichotomy conjecture for algebraically closed
fields (Conjecture \ref{RTC}) can be reformulated, roughly, as
follows:
\begin{conjecture}[Zilber's restricted trichotomy]
  Let $M$ be an algebraic curve over an algebraically closed field
  $k$. Let $X\subseteq M^2\times T$ be the total space of an ample
  family of plane curves, then $k$ can be reconstructed from the
  incidence system associated with $X$.
\end{conjecture}
In the above statement $X$ is ample\footnote{The terminology is
  borrowed with a twist from \cite{hz}. A family of curves is ample if
  it witnesses ampleness of the associated Zariski structure, in the
  sense of \cite{hz}.} if for any $(P,Q)$ in a dense open subset
$U\subseteq M^2\times M^2$ there exists a curve $X_t$ incident to both
$P$ and $Q$. In precise technical terms, the reconstruction of the
field is first order interpretability. We define this notion in the
next sub-section but for all practical purposes it suffices to know
that by standard model theoretic techniques described in Section
\ref{configurations} the problem reduces to the construction of a
certain combinatorial configuration, known as Hrushovski's field
configuration. Though more abstract than Desargues' or Pappus'
configurations, in practice, the field configuration is constructed in
this paper by finding a specific configuration of curves in $M^2$
satisfying certain incidence requirements (reflecting certain
tangencies between them).

In \cite[\S 2.4]{ArtinGeometry} not only is the field recovered from
the affine geometry, but also the geometry is recovered as the affine
plane over that field. In the present setting, there are examples due
to Hrushovski (see, e.g., \cite{Martin}) showing that the full Zariski
structure on $\mathfrak M$ cannot be recovered from $\Mm$. This can
probably be achieved if $X$ is \emph{very ample} (in the sense of
\cite{hz}, namely that $X$ separates points in $M^2$), but we do not
study this question here.

\subsection{Basic model theoretic notions}
In the previous sub-section we introduced the structure $\Mm$ viewed
as an incidence system. Since Zilber's full trichotomy conjecture is
false, in order to prove it in the present context, we have to exploit
the fact that $\mathfrak M$ is an algebraic curve, and that $X\to T$
is an algebraic family of algebraic curves. However, the abstract
model theoretic setting (allowing the curves in $X$ not to be proper,
irreducible smooth or pure-dimensional) is not well suited for
algebro-geometric analysis. In order to be able to address this
problem (and in order to apply the model theoretic machinery allowing
us to do so) we may have to change $X$ (e.g., replace it with a larger
family, or with a family whose members are pure dimensional). This has
to be done, of course, within the framework of the structure $\Mm$. In
model theoretic terms, the new family $X$ has to be \emph{definable}
in $\Mm$.

Formally, a set $C\subseteq M^n$ is \emph{$\0$-definable} in $\Mm$ if
it is contained in the smallest collection of subsets of
$\bigcup_{k} M^k$ containing all diagonals, the total space of $X$,
and which is closed under finite Boolean combinations, projections and
finite Cartesian products. The set $C$ is \emph{$\Mm$-definable} if
there exists a $\0$-definable $S\to R$ and some $r\in R$ such that
$C=S_r$. In practice, however, most definable sets we will be using
are obtained by a few simple constructs: the restriction of $X$ to
sub-families passing through a fixed point $P\in M^2$, intersections,
and the composition $X\circ X$ (see Section \ref{sec:composition}).

Another important means of obtaining $\Mm$-definable sets in the
present paper is as follows: we fix a definable family of plane curves
(say, the original family $X$ we started with) and a fixed
$\Mm$-definable curve $\mathcal C\subseteq M^2$ (e.g., a curve $X_t$
from our family). Then the sets
$X(n):=\{s\in T: \#(X_s\cap \mathcal C)=n\}$ are definable for all
$n$. We point out that even if $T$ is reducible (in the sense of the
full Zariski structure) but stationary in the sense of $\Mm$ (see
below) there will be a (unique) $n\in \mathbb N$ such that $X(n)$ is
dense open.

The notion of definability over parameters was mentioned above only
implicitly (when taking fibres of $\0$-definable sets). Due to its
importance for model theoretic notions appearing in the text we return
to it from a slightly different angle. Recall that we are working in
the full Zariski structure $\mathfrak M$ on a curve defined over an
algebraically closed field $k$, and an incidence system $\Mm:=(M,X)$
where $X$ is the total space of an ample family of algebraic curves in
$M^2$. This construction can be carried out over any (algebraically
closed) $K\ge k$. In $\mathfrak M(K)$ -- the full Zariski structure on
$\mathfrak M$ viewed as a curve over $K$ -- some of the closed sets
are extensions of closed sets in $\mathfrak M(k)$, and we refer to
them as $k$-definable, and some are new (e.g., they have no
$k$-rational points). Thus, a set $S$ is $\Mm$-definable over $K$ if
it is $\0$-definable in $\Mm(K)$.

To sets definable in $\Mm(K)$ (any algebraically closed field
$K\ge k$) we associate, naturally, a notion of dimension (the Krull
dimension of the Zariski-closure of the set in the full Zariski
structure, $\mathfrak M(K)$). We will refer to this notion of
dimension as \emph{Morley rank}. In those terms $M$ has Morley rank 1.
A definable set is \emph{stationary} or $\emph{$\Mm$-irreducible}$ if
it cannot be written as the disjoint union of two definable sets of
the same Morley rank.  Note that whereas the dimension of
$\Mm$-definable sets, when computed in $\mathfrak M$ and in $\Mm$ is
the same, the property of being stationary in the sense of $\Mm$ and
in the sense of $\mathfrak M$ need not be the same.

It follows from general model theoretic principles that there exists
$\Mm$-definable, $\Mm$-stationary $M_0\subseteq M$ of Morley rank 1,
such that $X_0:=X\cap M_0^2\times T$ is ample and, in fact $X_0$ can
be parametrised over an appropriate power of $M_0$. There is,
therefore, no harm in assuming that $M_0=M$, i.e., that $M$ is
$\Mm$-irreducible. A stationary definable set of Morley rank 1 is
\emph{strongly minimal}. Using (model theoretic) compactness, we may
also assume -- by a similar argument -- that for a dense open subset
$U\subseteq T$ (not, a priori, $\Mm$-definable) $X_t$ is strongly
minimal for all $t\in U$. A strongly minimal set is \emph{non locally
  modular} if it admits a definable ample family of plane curves.

Given a definable family $X\to T$ of curves in $M^2$ we define an
equivalence relation $\nsim$ on $T$ by $t\sim s$ if the Morley rank of
$X_t\triangle X_s$ is $0$, i.e., if the symmetric difference of $X_s$
and $X_t$ is finite. By general principles (known technically as
\emph{definability of Morley rank, or -- in this context -- the
  simpler elimination of the $\exists^\infty$ quantifier)}, this
equivalence relation $\sim$ is $\Mm$-definable if $X$ is, and by
strong minimality of the curves in $X$ it is, indeed, an equivalence
relation (on an $\Mm$-definable dense open subset of $T\times T$). The
family $X$ of plane curves is called \emph{faithful} (or
\emph{normal}) if $\sim$ is trivial. On general grounds we may assume,
without any further restrictions, that all classes of $\sim$ are
finite. By allowing so called \emph{imaginary elements} we may, in
fact, assume that $X$ is faithful. We will always assume that $X$ is
faithful\footnote{Our arguments would have worked, essentially,
  unaltered assuming only that $\nsim$ has finite classes. This would
  further overload the exposition with trivial technicalities we
  preferred to avoid.}, hiding under the rug our usage of imaginaries
(who will only play a short -- but important -- role in Section
\ref{cleanup-ind}).

In this terminology our goal is to show that, allowing imaginary
elements, we can find an infinite definable set $F$, and definable
function $\oplus,\otimes: F^2\to F$ (i.e., functions whose graphs are
definable sets) making $F$ into a field. To spell this out explicitly,
without using imaginary elements, we have to find an infinite
definable set, and definable functions
$\tilde\oplus, \tilde \otimes: S^2\to S$ and a definable equivalence
relation $E$ on $S$ such that $S/E$ equipped with $\tilde \oplus/E$
and $\tilde \otimes/E$ is an infinite field. By \cite[Theorem
4.15]{poizat} the resulting field is, necessarily, definably
isomorphic to $K$.


\subsection{Some standard reductions}
\label{section:setup}

As explained in Section \ref{intro-RTC}, the proof of Conjecture
\ref{RTC} can be reduced to the existence of a so called \emph{field
  configuration} (see Section \ref{configurations} for the details). A
key feature of field configurations is that they are stable under
definble finite-finite correspondences. In the present setting this
can be formulated as follows:  

\begin{fact}
  Let $D$ be a strongly minimal set (definable in some structure
  $\mathcal D$) and $R: D\to E$ a $\mathcal D$-definable finite-finite
  correspondence. Let $f$ be a field configuration in $D$ and $F$ the
  locus (in the full Zariski structure) of $f$. Then there exists
  $U\subseteq F$ dense and open such that $R(f')$ is a field
  configuration in $E$ for all $f'\in U$.
\end{fact}

It follows immediately from the definitions that if $M'$ is the
regular locus of $M$ and $X'$ is the restriction of $X$ to
$M'\times M'$ then $\Mm':=(M',X')$ is non-locally modular, and by the
above fact any field configuration in $\Mm'$ gives rise to a field
configuration in $\Mm$. This shows that we may assume, without loss of
generality, that $M$ is a regular curve.

If $R$ in the above fact is an $\Mm$-definable surjection then $E$ is
a quotient of $D$ by an equivalence relation with finite classes. The
converse is a little more delicate: if $\sim$ is an $\Mm$-definable
equivalence relation with finite classes then the quotient $M/\nsim$
need not be $\Mm$-definable (in technical terms, it is
\emph{interpretable}). It is, however, well known (see, e.g.,
\cite[Theorem 3.6]{HollyEOI} for an explicit construction) that the
quotient space is, in fact, an algebraic curve, therefore, if needed,
there is no harm replacing $M$ with $M/\nsim$. This will be applied in
Section \ref{second-group} as follows: in Section \ref{first-group} we
show that a strongly minimal group, $\mathcal G$, is interpretable in
$\Mm$ (it need not, in general, be definable); replacing $\Mm$ with
$\mathcal G$, we can then proceed to the construction of the field
configuration assuming that $\mathfrak M$ is a group, and that the
group operation is \emph{definable} in $\Mm$.

A slightly more delicate application of the same argument appears in
Section \ref{sec:ind} where we need our ample family $X$ to be
\emph{very ample}, which cannot be assured without passing to a
quotient. Thus throughout from now on we will be working under the
following assumptions:

\noindent\textbf{We fix a regular algebraic curve $M$ over an
  algebraically closed field $k$. We let $\mathfrak M$ denote the full
  Zariski structure on the curve. We fix a faithful ample family $X\to
  T$ of curves in $\mathfrak M\times \mathfrak M$ and denote
  $\Mm:=(M,X)$.} 


\section{Slopes}
\label{slopes}

The main object of study in the present section is slopes (of various
orders) of a curve at a regular point. While the notions we are
interested in make sense in a wider context, we only study them in the
context of curves in Cartesian powers $M^n$ of an algebraic curve over
an algebraically closed field. We introduce the operation of
composition (Subsection~\ref{sec:composition}) and taking sum of curves in
the presence of an underlying group structure
(Subsection~\ref{sum-of-correspondences}).

First, we adopt the convention to call \emph{curve} a variety of
dimension 1, not necessarily regular or irreducible; when we talk
about curves in $M^n$, we assume that they are Zariski closed in
$M^n$. A \emph{definable curve} is a definable set in $M^n$, of
dimension 1, not necessarily pure-dimensional or Zariski closed where
$(M,\ldots)$ is a reduct of the full Zariski structure on an algebraic
curve $M$.

\begin{defn}[Local coordinate system]
  \label{loc-coord} Let $P$ be a regular point of a variety $X$ of
  dimension $n$.  A \emph{local coordinate system at $P$} is an
  isomorphism $\widehat{\Oo_{X,P}} \xrightarrow{\sim} k[[x_1, \ldots,
  x_n]]$.
\end{defn}

By Corollary to Theorem 30.5 \cite{matsumura1989commutative} any
variety over a field is generically regular, and by Theorem~29.7
\emph{loc.cit.} a completion of a regular local ring over a field is a
formal power series ring.

If $X$ is one-dimensional and a local coordinate system is chosen at
$P$ then for all $n>0$ the inclusion $\Oo_{X,x} \to
\widehat{\Oo_{X,x}}$ followed by the reduction maps $k[[x]] \to
k[[x]]/(x^{n+1}) \cong k[x]/(x^{n+1})$ gives rise to closed embeddings
$\Spec k[x]/(x^{n+1}) \hookrightarrow M$ with the closed point of
$\Spec k[x]/(x^{n+1})$ mapped to $P$.

If $i_1: \widehat{\Oo_{X,P_1}} \xrightarrow{\sim} k[[x]]$, $i_2:
\widehat{\Oo_{Y,P_2}} \xrightarrow{\sim} k[[y]]$ are local coordinate
systems at regular points $P_1 \in X$, $P_2 \in Y$ then there is a
natural local coordinate system at the point $(P_1, P_2) \in X \times
Y$,
\[i_1 \otimes i_2: \widehat{\Oo_{X,P_1} \otimes \Oo_{Y,P_2}} \to
\varprojlim (k[x]/(x^n) \otimes k[y]/(y^n)) \cong k[[x,y]].\]

\begin{rem}
  From now on, whenever we mention that we choose a local
  coordinate system at $(P_1,P_2) \in M^2$ for $M$ a curve, we
  do so by choosing local coordinate systems $i_1, i_2$ at $P_1,
  P_2 \in M^2$ and then passing to $i_1 \otimes i_2$.
\end{rem}

Finally, let $f: X \to Y$ be a morphism of schemes. Recall
(\citep{ega} EGA IV.17.1) that a morphism $f$ is \emph{formally
\'etale} if for any scheme $T$, closed subscheme $T'$ defined by a
nilpotent ideal and any two compatible morphisms $\lambda: T' \to X$
and $\iota: T \to Y$ there is a unique morphism $\bar\iota: T \to X$
such that the following diagram commutes
$$
\xymatrix{
  T' \ar[r]^\lambda \ar[d] & X \ar[d]^{f}\\ T \ar[r]^\iota
  \ar@{-->}[ur]^{\bar\iota} & Y\\
}
$$
Observe that a closed embedding $\iota$ lifts to a closed embedding
$\bar\iota$ and it follows automatically that $f$ induces an
isomorphism between two copies of $T$ embedded into $X$ and $Y$.

A morphism $f: X \to Y$ is \emph{\'etale} if it is flat and
unramified. An \'etale morphism of schemes $f: X \to Y$ is formally \'etale (SGA1
\citep{sga1}, Corollaire~I.5.6), and a formally \'etale morphism which
is locally of finite presentation is \'etale (\citep{ega} EGA IV,
Corollaire~17.6.2). Since in what follows we deal mainly with schemes
of finite type over fields, we will not distinguish between these two
notions.

\begin{lemma}
  \label{sections} Let $f: X \to Y$ be an \'etale morphism.  Assume
  $Y$ is the spectrum of a local Artinian algebra over a field. Then
  sections of $f$ are in one-to-one correspondence with closed points
  in the fibre over the closed point of $Y$. For any such section $s$
  the composition $f \circ s$ is an isomorphism.
\end{lemma}

\begin{proof}
  By definition of formally \'etale morphism, given any morphism
  $\lambda: Y'\to X$ where $Y'$ is a reduced subscheme supported at a
  closed point of $X$ yields a unique section $s$ that makes the
  following diagram commute:
  $$
  \xymatrix{
    Y' \ar[r]^\lambda \ar[d] & X \ar[d]^{f}\\ Y \ar[r]^{id}
    \ar@{-->}[ur]^{s} & Y\\
  }
  $$

\noindent That $s \circ f$ is an isomorphism follows from
Corollary~I.5.3,
  SGA1~\citep{sga1}.
\end{proof}


\begin{defn}[Scheme-theoretic image]
  Let $f: X \to Y$ be a morphism of schemes. The \emph{scheme
    theoretic image of $X$ in $Y$} is the smallest closed subscheme of
  $Y$ through which $f$ factors.
\end{defn}

\begin{lemma}
  \label{factors-inf} Let $\Oo_{X,x}$ be a local ring with residue
  field $k$ and let $f: \Oo_{X,x} \to k[\eps]/(\eps^{n+1})$ be a
  morphism. Then $f$ factors through $\Oo_{X, x}/\m^{n+1}$ where $\m$
  is the maximal ideal of $\Oo_{X, x}$.
\end{lemma}

\begin{proof}
  One observes easily that $f^{-1}(\mathfrak{p}) \subset \m^{n+1}$
  where $\mathfrak{p}$ is the radical ideal of $k[\eps]/(\eps^{n+1})$,
  since $\mathfrak{p}^{n+1}=0$. This implies the statement of the
  lemma.
\end{proof}

Let $M$ be a curve, and assume a local coordinate system is chosen at
a regular point $P=(P_1, P_2) \in M^2$. Let $Z\subseteq M\times M$ be
a curve such that the projection of $Z$ on the first factor $M$ is
\'etale in an open neighbourhood of $P$.  Consider the diagram
\begin{figure}[H]
\centerline{\xymatrix{ 
  \Oo_{M, P_2} \ar[r]^-{f_2} \ar[d] \ar[ddr]^g & \Oo_{M,
    P_2}/\m_2^{n+1} \ar[dd]^{\tau_n} \ar@{=}[r]^-{\sim} &
  k[\eta]/(\eta^{n+1})\\ 
  \Oo_{Z,P} \ar[rd]^{\gamma} & & \\
  \Oo_{M, P_1} \ar[r]^-{f_1} \ar[u] & \Oo_{M, P_1}/\m_1^{n+1}
  \ar@{=}[r]^-{\sim} & k[\eps]/(\eps^{n+1})\\ 
}}\caption{Slopes}\label{fig:slopes}
\end{figure}
where $\m_1$, $\m_2$ are the maxmial ideals of $\Oo_{M,P_1}$,
$\Oo_{M,P_2}$, respectively.  The isomorphisms on the right are
provided by the local coordinate systems at $P_1$ and $P_2$. The
morphism $\gamma$ is a lifting of $f_1$ that follows from etaleness of
$\Oo_{Z,P}$ over $\Oo_{M,P_1}$. The morphism $g$ is the composition of
the structure morphism of the $\Oo_{M,P_2}$-algebra
$\Oo_{Z,(P_1,P_2)}$ and $\gamma$. By Lemma~\ref{factors-inf}, $g$
factors through $\Oo_{M,P_2}/\m^{n+1}$. The morphism $\tau_n$ that
comes out of this factorization, can be regarded as an endomorphism of
$k[\eps]/(\eps^{n+1})$ after identifying $k[\eta]/(\eta^{n+1})$ with
$k[\eps]/(\eps^{n+1})$.

\begin{defn}[Slope]
  \label{slope-definition} Let $M$ be a curve, and let
  $Z \subset M \times M$ be a curve as above. The \emph{$n$-th order
    slope of $Z$ at $(P_1, P_2)$} is the endomorphism
  $\tau_n: k[\eps]/(\eps^{n+1}) \to k[\eps]/(\eps^{n+1})$ arising from
  the construction above. In general, we will denote the slope of $Z$
  at $P$, which is an element of $\End(k[\eps]/(\eps^{n+1})$, as
  $\tau_n(Z, P)$.
\end{defn}


Similarly, in the above setting, if $N$ is an algebraic variety over
$k$ and $Z \subset M^2 \times N$ a curve, $P=(P_1, P_2, P_3) \in Z$ a
regular point such that the projection $p_1$ on the first factor $M$
is \'etale in an open neighbourhood of $(P_1, P_2, P_3)$, we consider
the diagram in the Figure \ref{fig:slopes}, and consider the lifting
$\gamma$ of the morphism $f_1$ that exists by etaleness of $\Oo_{Z,P}$
over $\Oo_{M,P_1}$. By Lemma~\ref{factors-inf}, $g$ factors through
$\Oo_{M,P_2}/\m^{n+1}$. The morphism $\tau_n$ that comes out of this
factorization may be viewed as an endomorphism of
$k[\eps]/(\eps^{n+1})$ after one identifies $k[\eta]/(\eta^{n+1})$
with $k[\eps]/(\eps^{n+1})$.

\begin{defn}[Relative slope]
  \label{relslope-definition} The \emph{$n$-th order slope of $Z$ at
    $P$ relative to $N$}, denoted $\tau_n(Z/N, P)$, is the endomorphism
  $\tau_n: k[\eps]/(\eps^{n+1}) \to k[\eps]/(\eps^{n+1})$ arising from
  the construction above. 
\end{defn}

The need for this seemingly artificial definition will become apparent
in the next section; it serves the same purpose as the notion of
``branches of curves'' in \citep[Section~3.8]{zg}.

In the above definition, and later in this article, we tacitly assume
that a choice of local coordinate systems for $M$ at the relevant
points has been made. 

A first-order slope is a map $k[\eps]/(\eps^2) \to k[\eps]/(\eps^2)$,
determined by its action on $\epsilon$:
$\epsilon \mapsto a \cdot \epsilon$. We observe that the scalar $a$ is
just a component of the normalised Pl\"ucker coordinates of the
tangent subspace in the given local coordinate system. Clearly, two
curves having the same first order slope at a point are tangent at
this point.

\begin{defn}[Formal power series expansion]
  \label{formal-expansion} In the setting of
  Definition~\ref{slope-definition} consider the endomorphism
  $$
  A = \varprojlim_n \tau_n(Z, (P_1, P_2)) \in \varprojlim
  (\End(k[\eps]/(\eps^{n+1}) \cong \End(k[[x]])
  $$
We call $A(x)$ the \emph{formal power series expansion} of $Z$ at
$(P_1, P_2)$.  

Similarly, if $Z \subset M^2 \times N$, the relative formal power
series expansion of $Z$ at $Q \in Z$ is the result of applying the
endomorphism
$$
\varprojlim_n \tau_n(Z/N, Q)
$$
to $x \in k[[x]]$.
\end{defn}

The following is now immediate from the definitions; 

\begin{prop}
  \label{expansion-is-point}
  Given local coordinate systems at $P_1,P_2\in M$ there is a
  canonical isomorphism
  $\widehat{\Oo_{M^2, (P_1, P_2)}} \to k[[x,y]]$. The formal power
  series expansion of a curve $Z$ at $(P_1,P_2)$ is $f \in k[[\eps]]$
  if and only if the morphism $\Oo_{M^2,(P_1, P_2)} \to k[[\eps]]$
  given by $x \mapsto \eps$, $y \mapsto f$ factors through
  $\Oo_{Z,(P_1, P_2)}$. 

  Similarly, the relative formal power series expansion of a curve
  $W \subset M^2 \times N$ at $(P_1, P_2, P_3)$ is $f \in k[[\eps]$ if
  and only if the morphism $\Oo_{M^2,(P_1, P_2)} \to k[\eps]/(\eps^n)$
  given by $x \mapsto \eps$, $y \mapsto f \mod \eps^n$ factors through
  $p(\Oo_{W,(P_1, P_2, P_3)}/\m^n)$, where $p: M^2 \times N \to M^2$
  is the projection, for all $n > 1$.
\end{prop}

Later on we will need the following characterisation of invertible
endomorphisms in $\End(k[\eps]/(\eps/{n+1})$:

\begin{prop}
  \label{automorphisms} Let $\Aut( k[\eps]/(\eps^{n+1}))$ denote the
  set of automorphisms of $ k[\eps]/(\eps^{n+1})$.  Consider the
  restriction map
  $\End( k[\eps]/(\eps^{n+1})) \to \End( k[\eps]/(\eps^2) )$ defined by
  $\varphi\mapsto (f \mapsto \varphi(f)/(x^2))$. Then
  $\Aut( k[\eps]/(\eps^{n+1}))$ is the pre-image of
  $\Aut( k[\eps]/(\eps^2))$.
\end{prop}

\begin{proof}
  An endomorphism $\varphi\in \End( k[\eps]/(\eps^{n+1}))$ is
  invertible if and only if there exists $f$ such that
  $\varphi(f)=x$. By Corollary~7.17, \citep{eisenbud}, an endomorphism
  of $k[[x]]$ defined by sending $x$ to $f$ is an automorphism if and
  only if $f \in (x)$ but not $f \in (x^2)$. As any
  $\varphi\in \End( k[\eps]/(\eps^{n+1}))$ extends uniquely to
  $\End(k[[x]])$, the conclusion follows.
\end{proof}

\begin{defn}[Graph of a morphism]
  Let $f: X \to Y$ be a morphism of schemes over some base scheme
  $S$. The \emph{graph} of the morphism $f$ is the unique closed
  subscheme $Z \subset X \times_S Y$ such that the projection on $X$
  restricted to $Z$ is an isomorphism and $f = p_Y \circ p_X^{-1}$.
\end{defn}

We can now state: 

\begin{lemma}
  \label{slope-graph} 
  Let $Z \subset M \times M$ be a locally closed set of dimension 1
  such that the slope is well-defined at $P \in Z$. Let
  $\Oo_{M^2,P} \to k[\eps,\eta]/(\eps^{n+1}, \eta^{n+1})$ be the
  natural morphism induced by the local coordinate system.  Denote
  $$
  R := \Oo_{Z,P} \otimes_{\Oo_{M^2,P}}
  k[\eps,\eta]/(\eps^{n+1},\eta^{n+1}) \textrm{ and } \wbar{Z} :=
  \Spec R
  $$
  Then $\wbar{Z}$, identified with a closed subscheme of
  $\Spec k[\eps, \eta]/(\eps^{n+1}, \eta^{n+1})$, is the graph of
  $\tau_n(Z,P)$.  If $p_2: Z\to M$ is \'etale in an open neighbourhood
  of $P$ then it is is an automorphism of $k[\eps]/(\eps^{n+1})$.
\end{lemma}

\begin{proof}  
  Consider $R$ in relation to the objects in Figure \ref{fig:slopes}
  as per the diagram below. It follows from the definition that the
  ring $R$ has natural $\Oo_{M,P_1}/\m_1^{n+1}$-algebra and an
  $\Oo_{M,P_2}/\m_2^{n+1}$-algebra structures, giving rise to the
  morphisms in the diagram denoted, by a slight abuse of notation, by
  $p_1$ and $p_2$:
  $$
  \xymatrix{ 
     &  \Oo_{M, P_2} \ar[r]^-{f_2} \ar[d] \ar[ddr]^g &
     \ar@/_4pc/[dll]_{p_2} \Oo_{M, P_2}/\m_2^{n+1} \ar[dd]^{\tau_n} \ar@{=}[r]^-{\sim} & k[\eta]/(\eta^{n+1})\\
    R & \ar[l] \Oo_{Z,P} \ar[rd]^{\gamma} & & \\
    &  \Oo_{M, P_1} \ar[r]^-{f_1} \ar[u] & 
       \ar@/^4pc/[ull]^{p_1} \Oo_{M,P_1}/\m_1^{n+1} \ar@{=}[r]^-{\sim} & k[\eps]/(\eps^{n+1})\\ 
  }
  $$
  
  It follows from the fact that $\Oo_{Z,P}$ is \'etale over
  $\Oo_{M,P_1}$ that the morphism $p_1$ is an isomorphism, and if
  $\Oo_{Z,P}$ is \'etale over $\Oo_{M,P_2}$ then $p_2$ is an
  isomorphism too. The conclusion follows from the commutativity of
  the diagram.
\end{proof}

The above is also true for relative slopes. As we will repeatedly use the relative version of the lemma we state it explicitly. 

\begin{lemma}
  \label{relslope-proj} Let $Z \subset M^2 \times N$ be a locally
  closed one-dimensional set, $P\in Z$ such that the relative slope is
  defined at $P$. Let $p: Z \to M^2$ be the natural projection. Let
  $\Oo_{M^2,P} \to k[\eps,\eta]/(\eps^{n+1}, \eta^{n+1})$ be the
  natural morphism induced by the local coordinate system. Define
  $$
  R := \Oo_{Z,P} \otimes_{\Oo_{M^2\times N,P}}
  k[\eps,\eta]/(\eps^{n+1},\eta^{n+1}) \textrm{ and } \wbar{Z} :=
  \Spec R
  $$
  Then the scheme-theoretic image of $\wbar{Z}$ under $p$ is the
  graph of $\tau_n(Z/N,P)$.  If $p_2: Z\to M$ is \'etale in an open
  neighbourhood of $P$ then it is  an automorphism of
  $k[\eps]/(\eps^{n+1})$.
\end{lemma}

\begin{proof}
  Proceed as in the proof of the previous lemma to show the
  commutativity of the diagram. It is then left to show that $p$
  induces an isomorphism between $\wbar{Z}$ and its scheme-theoretic
  image under $p$. But this is immediate from the universal property
  of the scheme-theoretic image, and the fact that projections $p_1$ of
  both $\wbar{Z}$ and $p(\wbar{Z})$ on $M$ induce an isomorphism with
  one and the same closed subscheme of $M$.
\end{proof}



\subsection{Composition of curves}
\label{sec:composition} We have identified the slope of a curve at a
regular point with an endomorphism of $\Spec k[\eps]/(\eps^{n+1})$. We
note that the group of automorphisms of $\Spec k[\eps]/(\eps^2)$ is
$\G_m(k)$, and that in general an automorphism group of a fat point
$\Spec k[\eps]/(\eps^{n+1})$ is unipotent of rank $n$. Our goal in
Section~\ref{interp-field} will be to recover $\G_m(k)$ by identifying
its points with the (first order) slopes of a family of curves in
$M^2$.  In positive characteristic we will have to resort to higher
order slopes and we will not be able to assure that the group recovered
will indeed be $\G_m(k)$, but the general outline of the construction
remains the same. The group operation on slopes arises from the
operation of composition of curves. In this sub-section we
develop the necessary machinery.

We adopt the notational convention to denote by $p$ with subscripts
the projections from a product of several schemes or varieties on the
product of factors of the product, with the numbers of these fuctors
mentioned in the subscript. For example, $p_{124}$ on $M \times M
\times M \times N \times N'$ is the projection on the product $M
\times M \times N$ where the two $M$s are the first and second
factors $M$.

By way of motivation, let us consider first the following operation
on pairs of curves in $M^2$, regarded as correspondences. 

\begin{defn}[Composition]
  \label{defn:composition}
\begin{enumerate}
\item Let $X, Y$ be two curves in $M^2$, cut out by ideal sheaves
  $I_X$, $I_Y$ respectively. Their \emph{composition}, denoted
  $Y \circ X$, is the scheme cut out by the ideal sheaf
  $$
  (p_{13})_*(p_{12}^* I_X \otimes p_{23}^* I_Y).
  $$
\item If $X$ and $Y$ are one-dimensional
  definable subsets of $M^2$ then define their composition $Y \circ X$
  to be the definable set
  $$
  Y \circ X := \suchthat{ (x,z) \in M^2} { \exists y\ \ (x,y) \in X
    \textrm{ and } (y,z) \in X}
  $$
\end{enumerate}
\end{defn} 

Our scheme theoretic composition is, by definition, the
scheme-theoretic image under $p_{13}$ of the scheme
$X \times_{p_2, p_1} Y$, naturally embedded into $M^3$. One can show
that the if $X$ and $Y$ are graphs of functions $f,g: M \to M$, then
their composition is the graph of the function $f \circ g: M \to
M$. This definition coincides with the definable version on points in
case $X$, $Y$ are closed.

One can show that
\begin{equation}
\tau_n(Y \circ X, (P_1, P_3)) = \tau_n(X, (P_1, P_2)) \tau_n(Y, (P_2,
P_3))\label{eq:slope-mult}
\end{equation}
if all the slopes occurring in the equation are
well-defined. Unfortunately, even if the slope is defined for $X$ at
$(P_1, P_2)$ and for $Y$ at $(P_2, P_3)$, the composition $Y \circ X$
may be not regular at $(P_1, P_3)$.

Moreover, when we reconstructing the groups and fields from $\Mm$ for
a curve $M$ defined over a field of positive characteristic we have to
work with curves embedded into $M^n$ for $n > 2$.

We generalise, therefore, Definition~\ref{defn:composition} in two
directions: firstly, to apply to curves embedded into $M^n$, $n>2$,
secondly, to produce curves that are known to have a well-defined
relative slope at a certain point so as to make an analogue of
equation (\ref{eq:slope-mult}) well-defined.

\begin{defn}[Relative composition]
  \label{rel-composition-def} Let $M$ be a curve over an algebraically
  closed field, $N, N'$ some varieties, and let
  $X \subset M^2 \times N, Y \subset M^2 \times N'$ be curves. The
  \emph{relative composition $Y \circ X$ of $X$ and $Y$} is the
  1-dimensional subscheme of $M^3 \times N \times N'$ cut out by the
  ideal sheaf
  $$
  p_{124}^* I_X \otimes p_{235}^* I_Y. 
  $$

  If $X, Y$ as above are definable the relative composition of $X$ and $Y$ is 
  \[
  \suchthat{ (x,y,z,u,v) \in M^3 \times N \times N'}{ (x,y,u) \in X
    \textrm{ and } (y,z,v) \in Y }.
	\]
\end{defn}

% remark: we use the same notation, but relative composition does not
% make this final projection. so if we apply relative composition to
% actual curves in M^2 (times a point), this is not the same as
% applying the usual composition. Luckily, we never do 

In order to relate the relative composition with the composition of
slopes, we need the following version of the projection formula: 

\begin{lemma}
\label{proj-formula} Let $p: X \to Y$ be an affine morphism of
schemes, and let $\Ee, \Ff$ be coherent sheaves on $X,Y$
respectively. Then the natural morphism
$$
p_* \Ee \otimes \Ff \to p_*(\Ee \otimes p^* \Ff)
$$
is an isomorphism.
\end{lemma}

\begin{proof}
  Follows from Corollaire~I.9.3.9, \citep{ega1new}.
\end{proof}

\begin{prop}
  \label{rel-composition}
  Let $X$ and $Y$ be as in the Definition~\ref{rel-composition-def},
  $P_1, P_2, P_3\in M$, $P_4\in N$ and $P_5\in N'$ be such that the slope of $X$ is
  defined at $(P_1, P_2, P_4)$, and the slope of $Y$ is
  defined at $(P_2, P_3, P_5)$ with respect to some fixed local
  coordinate systems. Then for any $n > 0$
  $$
  \begin{array}{rcl}
  \tau_n(Y \circ X/M \times N \times N', & (P_1, P_2, P_3, P_4, P_5))
                                            =  &\\
  &\multicolumn{2}{l}{ = \tau_n(X/N, (P_1, P_2, P_4))\cdot
  \tau_n(Y/N', (P_2, P_3, P_5))} \\
  \end{array}
  $$
  \noindent where the relative slope in the left hand side is relative
  to the product of the second factor $M$ in $M^3$ and $N \times N'$.
\end{prop}

\begin{proof}
  Denote $T = \Spec k[\eps]/(\eps^{n+1})$.  Cartesian powers of $T$
  embed into Cartesian powers of $M$ according to the chosen local
  coordinate systems.
  
  By Lemma~\ref{relslope-proj}, for all $n>0$,
  $$
  \Gamma(\tau_n(X/N, (P_1,P_2, P_4))) \cong p_{12}(X \times_{M^2} T^2)
  $$
  and
  $$
  \Gamma(\tau_n(Y/N', (P_2,P_3,P_5))) \cong p_{23}(Y \times_{M^2} T^2)
  $$
  Let $I_X, I_Y$ denote the ideal sheaves that cut out
  $X \times_{M^2} T^2$, $Y \times_{M^2} T^2$ respectively in 
  $M^2 \times N$ and $M^2 \times N'$.

  Let $f$ and $g$ be the endomorphisms of $T$ associated with
  $\tau_n(X/N, (P_1,P_2,P_4))$ and $\tau_n(Y/N', (P_2,P_3,P_5))$. Let
  $\gamma_X: T \to T \times T \times N$, $\gamma_Y: T \to T \times T
  \times N'$ be the sections of the projections onto the first factor $T$ of 
  $X \times_{M^2} T^2 \subset T \times T \times N$ and 
  $Y \times_{M^2} T^2 \subset T \times T \times N'$ respectively. Let
  $\eta_Y: T \times T \times N \to T \times T \times T \times N \times
  N'$ be the map acting as the identity on the first $T$ and on $N$
  and as $\gamma_Y$ on the second $T$ but sending the result to the
  product of the second, third and fifth factors of the destination. That is, 
  \[
  \eta_Y=p_1\times P_1\circ \gamma_y \circ p_2 \times P_2\circ
  \gamma_y \circ p_2 \times p_3 \times P_3\circ \gamma_y \circ p_2 
  \]
  where the $p_i$ denote the projections on $T\times T\times N$ and
  $P_i$ the projections on $T\times T\times N'$.  
  
  One gets, immediately from these definitions, that
  \begin{equation*}
    p_1 \circ \gamma_X = f \qquad 
    p_{124} \circ \eta_Y = \id
    \label{eq:identities}    
  \end{equation*}
  and that $p_{235}^* I_Y \cong \eta_{Y*} \Oo_{T \times T \times N}$, and
  that $I_X \cong \gamma_{X*} \Oo_T$.

  It follows from the definition of relative composition and
  Lemma~\ref{relslope-proj} that
  $$
  p_{13*}(p_{124}^* I_X \otimes p_{235}^* I_Y)
  $$
  is the ideal that cuts out the graph of
  $\tau_n(Y \circ X/M \times N\times N', (P_1, P_2, P_3, P_4, P_5))$
  in $T \times T$.

  
  Then by Lemma~\ref{proj-formula} (which we can apply because the map
  $\eta_Y$ is affine and all the sheaves involved are coherent),
  $$
  \begin{array}{rl}
    & p_{13*} (p_{124}^* I_X \otimes p_{235}^* I_Y) = \\
    = & p_{13*} ( p_{124}^* I_X \otimes \eta_{Y*} \Oo_{T \times T \times N} )  = \\
    = &  p_{13*} \eta_{Y*}  \eta_Y^* p_{124}^* \gamma_{X*} \Oo_T = \\
    = &  (p_{13} \circ \eta_Y)^*  (p_{124} \circ \eta_Y)^* \gamma_{X*}
        \Oo_T = \\
    = &  (p_{13} \circ \eta_Y \circ \gamma_{X*}) \Oo_T\\
  \end{array}
  $$

  The composition $p_3 \circ \eta_Y \circ \gamma_{X*}: T \to T$ is the
  morpmism $g \circ f$,: indeed, in the diagram
  $$
  \xymatrix{
    T \ar[r]^-{\gamma_X} \ar[dr]^{f} & T \times T \times N
    \ar[r]^-{\eta_Y} \ar[d]^{p_2} \ar[dr]^{g \circ p_2} & T
    \times T \times T \times N \times N' \ar[d]^{p_3} \\
    & T \ar[r]^g & T\
  }
  $$
  the commutativity of the upper left triangle follows from the
  definition of $\gamma_X$, and commutativity of the upper right
  triangle follows from the definiton of $\eta_Y$.

  We have shown, therefore, that $p_{13*}(p_{124}^* I_X \otimes p_{235}^* I_Y)$ is
  the ideal $(\id \times (g \circ f))_* \Oo_T$ which is, by definition,the
  ideal that cuts out the graph of $g \circ f$ in $T \times T$.
\end{proof}

Note that the above proposition can be applied to curves in $M^2$ by taking $N,N'$ to
be points.

Recall that formal power series can be composed in the following
sense: given an element $y \in k[[x]]$ from the maximal ideal, there
exists a unique homomorphism of topological rings
$f_y: k[[x]] \to k[[x]]$ mapping $x$ to $y$ (see \citep{algebre},
IV,\S 3 for example). The image of a power series $z \in k[[x]]$ under
$f_y$ is the composition of the power series $y$ with the power series
$z$. Since the expansion of a curve $Z$ at a regular point is defined
through the inverse system of $n$-slopes, it is then easy to verify
that, by of Proposition~\ref{rel-composition}, the formal power series
expansion or $W\circ V$ is the composition of the formal power series
expansion of $V$ with the power series expansion of $W$, expansions
taken at appropriate points. This allows us to carry out the following
useful manipulation:

\begin{lemma}
  \label{cancel-out-pth-power} Fix local coordinate systems at
  $P_1,P_2, P_3\in M$. Let $Z_1,Z_2\subseteq M^2$ be curves with
  $(P_2,P_1)\in Z_1$ and $(P_2,P_3)\in Z_2$ both regular points on the
  respective curves. Let $f_1, f_2 \in k[[x]]$ be the associated power
  series expansions of $Z_1,Z_2$ at $(P_2,P_1)$ and at $(P_2,P_3)$
  respectively. Assume that $f_1 = g_1^{p^n}, f_2 = g_2^{p^m}$ with
  $g_1, g_2 \in k[[x]]$ and $g_1$ such that the automorphism of
  $k[[x]]$ sending $x$ to $g_1$ is invertible, with the inverse
  automorphism sending $x$ to $g_1'$. Then the (relative) power series
  expansion of the (relative) composition $Z_2 \circ Z_1^{-1}$ at
  $(P_1,P_3)$ (respectively, $(P_1, P_2, P_3)$) is given by
  $h: = (g_2 \circ g_1')^{p^{m-n}}$.
\end{lemma}

\begin{proof}
  First let us prove the statement for $M=\A^1$ and
  $P_1=P_2=P_3=0$. Since $Z_1$, $Z_2$ are regular at $(0,0)$ only one
  irreducible component of each of $Z_1$, $Z_2$ passes through
  $(0,0)$, so we may assume that $Z_1$ and $Z_2$ are irreducible. In
  that case it is enough to notice that if a formal power series
  expansion of a curve at $(0,0)$ is of the form $f^{p^n}$ then this
  curve is a composition (in the sense of
  Section~\ref{sec:composition}) of some curve with formal power
  series expansion $f$ and the graph of the $n$-th power of the
  Frobenius morphism $\A^1 \to \A^1$, and the necessary statement
  follows.
  
  For the general case, find projections $p_i: M \to \A^1$ that map
  $P_i$ to $0$ and such that $p_i$ is \'etale in a neighbourhood of
  $P_i$. Since \'etale morphisms induce isomorphisms of completed
  local coordinate rings, local coordinate systems at $P_1, P_2, P_3$
  are induced by precomposing with $p_1, p_2, p_3$
  respectively. Observe that the slope of $Z_1$ at $(P_2, P_1)$
  coincides with the slope of $(p_2 \times p_1)(Z_1)$ and similarly
  for $Z_2$ and $(p_2 \times p_3)(Z_2)$. Then the statement follows
  from the statement for $\A^1$ which we have already
  proved. 
\end{proof}




\subsection{Sum of curves}
\label{sum-of-correspondences}

We are now going to define another binary operation on curves in $M^2$
and more generally in $M^n$, $n>2$, making use of an algebraic group
structure on $M$. This operation amounts to the application of the
group law to the second coordinate of points in $M^2$ and we call it,
abusing terminology and notation, ``the sum of curves''.  As
the operation of composition, it can be defined in a compatible way
scheme theoretically. When applied to graphs of maps from the double
point to an algebraic group, the operation amounts to addition of
tangent vectors which can be taken as a justification for the
terminology. Ultimately, this operation will allow us to recover,in
Section~\ref{second-group}, the additive group of the field and the
action of the multiplicative group on it.

\begin{defn}[Sum of definable curves in $M^2$]
\label{sum-definable}
  Let $(G,+)$ be a definable group and let
  $X,Y \subset G \times G$ be definable curves. Define:
  $$
  {\setlength\arraycolsep{2pt}
  \begin{array}{lr} X  + Y := \{(x,u) \in G \times G \mid
  \exists y,z\in G\ \ (x,y) \in X & \textrm{and } (x,z)
  \in Y, \\
     & \textrm{and } u = y+z)\}\\
  \end{array}}
  $$
\end{defn}

\begin{rem}
  The notation above suggests that $G$ is commutative. The definition
  applies even if it is not the case, although in this paper we will
  only deal with $G$ which are Abelian.
\end{rem}

Note that the sum of curves does not in general define a
group law on the class of all definable curves in $G^2$: in fact, the
obvious candidate for a curve ``opposite'' to a
curve $X$,
$$
-X  := \suchthat{ (x,-y) }{ (x,y) \in X}
$$
is an opposite with respect to ``+'' only if $X$ is a graph of a
function.

The compatible scheme-theoretic operation is defined as follows: 

\begin{defn}[Scheme-theoretic sum of curves]
  \label{corresp-sum} Let $G$ be an algebraic group,
  $X,Y \subset G \times G$ curves.  Define $X + Y$ to be the curve cut
  out in $G \times G$ by the ideal
  $$
  I_{X + Y} = (p_1 \times a\circ p_{23})_* (p_{12}^* I_X \otimes p_{13}^* I_Y)
  $$
  where $a: G \times G \to G$ is the group law morphism and $p_1, p_{ij}$ are projections on $G^3$.
\end{defn}

Guided by the same considerations that motivated the introduction of
the relative composition (Definition~\ref{rel-composition-def}), we
define an operation on curves in $G^n$ generalising that of
Definition~\ref{corresp-sum} and \ref{sum-definable}, with the aim of
expressing the relative slope of the result of this operation as a sum
of the relative slopes of its arguments.

\begin{defn}[Relative sum]
\label{relsum-defn}
Let $G$ be a 1-dimensional algebraic group, $N, N'$ algebraic
varieties, and $X \subset G^2 \times N$, $Y \subset G^2 \times N'$
curves. We define the \emph{relative sum $X + Y$ of $X$ and $Y$} to be
the curve cut out by the ideal
$$
I_{X + Y} = (p_1\times a \circ p_{24})_* (p_{123}^* I_X \otimes
p_{145}^* I_Y)
$$
where $a: G \times G \to G$ is the group law morphism and
$p_{ij},p_{ijk}$ are projections from
$G^2\times N\times G\times N'$ on products of factors of
$G^3 \times N \times N'$.

If $X$ and $Y$ are definable sets, their relative sum is defined on
as follows:
$$
X + Y := \{ (a,b,c, d, e,b+d) \mid (a,b,c) \in X, (a,d,e) \in Y \}
$$    
\end{defn}

 % \begin{prop}
 %   \label{p1-points-are-ga} Let $G$ be an algebraic group over a
 %   field $k$ and let $\rho: G(\Spec k[\eps]/\eps^2) \to G(k)$ be the
 %   natural restriction map. Then $\Ker \rho$ is isomorphic to
 %   $\G_a^n(k)$ where $n=\dim G$.
 % \end{prop}

 % \begin{proof}
 %   The set $\Ker \rho$ is naturally identified with the tangent space
 %   $T_0G$, and the differential of the group law is some bi-linear map
 %   $\bar a: T_0 G \otimes T_0 G \to T_0 G$. It follows from
 %   bi-linearity that
 %   $$
 %   \bar a (x \otimes y) = Ax + By
 %   $$
 %   for some $A,B \in GL_n(k)$.

 %   It follows from associativity of the group law that:
 %   $$
 %   A(Ax + By) + Bz = Ax + B(Ay+Bz)
 %   $$
 %   for all $x,y,z \in T_0 G$. From that we easily deduce that $A^2=A,
 %   B^2=B$. Therefore necessarily, $A$ and $B$ are the identity maps,
 %   and $a(x,y)= x+y$.
 % \end{proof}

 Let $G$ be a one-dimensional algebraic group, then the \emph{formal
   group law of $G$} is defined as the image of the topological
 generator of $k[[x]]\cong \widehat{\Oo_{G,e}}$ under the morhpism
 $\widehat{\Oo_{G,e}} \to \widehat{\Oo_{G,e}} \hat{\otimes}
 \widehat{\Oo_{G,e}} \cong k[[x,y]]$ induced by the group operation
morhpism. It is a well-known fact that the lower order terms of a
one-dimensional formal group law are $x + y$.

If $F \in k[[x,y]]$ is a one-dimensional formal group law, we denote
$F_n$ its $n$-th order truncation in $k[x,y]/(x^{n+1},y^{n+1})$.

 \begin{prop}
   \label{addition} Let $G$ be a 1-dimensional algebraic group over an algebraically
   closed field $k$ with the group law morphism $a: G \times G \to G$. 

   Let $X \subset G \times G \times N$, $Y \subset G \times G \times N'$
   be curves and assume that slopes are defined at the points
   $(P_1,P_2,P_3) \in X$ and $(P_1, P_4, P_5) \in Y$. Then for all $n\geq 1$
   $$
   \begin{array}{rcl}
   \tau_n(X + Y/G^2 \times N \times N', & (P_1,P_2,P_3,P_4,P_5,P_2+P_4)
      =  &\\
     & \multicolumn{2}{l}{ = F_n(\tau_n(X/N, (P_1,P_2,P_3)),\tau_n(Y/N',(P_1,P_4,P_5)))}\\
   \end{array}
  $$  
  where $F$ is the formal group law of $G$, and $F_n$ is viewed as a
  morphism from  $k[\eps]/(\eps^{n+1})$ to $k[x,y]/(x^{n+1},y^{n+1})$
  sending $\eps$ to $F_n(x,y)$.  In particular, for $n=1$,
   $$
   \begin{array}{rcl}
   \tau_1(X + Y/G^2 \times N \times N', & (P_1,P_2,P_3,P_4,P_5,P_2+P_4)
      =  &\\
     & \multicolumn{2}{l}{ = \tau_1(X/N, (P_1,P_2,P_3)) + \tau_1(Y/N',(P_1,P_4,P_5))}\\
   \end{array}
  $$  
\end{prop}

\begin{proof}
  The setting is as in the proof Proposition~\ref{rel-composition}, so we review the notation briefly: we set
%  As in Proposition~\ref{rel-composition},
%  Cartesian powers of $T$ embed into Cartesian powers of $G$ according
%  to the local coordinate systems chosen.
%  By Lemma~\ref{relslope-proj}, for all $n \geq 0$,
%  $$
%  \Gamma(\tau_n(X/N, (P_1,P_2,P_3))) \cong p_{12}(X \times_{G^2} T^2) 
%  $$
%  and
%  $$
%  \Gamma(\tau_n(Y/N', (P_1,P_4,P_5))) \cong p_{12}(Y \times_{G^2} T^2) 
%  $$
	\begin{itemize}
		\item $T = \Spec k[\eps]/(\eps^{n+1})$,
		\item $I_X,I_Y$ the ideal sheaves cutting out $X \times_{G^2}
		T^2$, $Y \times_{G^2} T^2$, respectively in $G^2 \times N$,
		and in $G^2 \times N'$
		\item $f,g$ be the endomorphisms of $T$ associated with
		$\tau_n(X/N, (P_1,P_2,P_3))$ and $\tau_n(Y/N', (P_1,P_4,P_5))$.
		\item $\gamma_X: T \to T \times T \times N, \gamma_Y: T \to T \times T \times N'$ are the sections of the projection
		$X \times_{M^2} T^2 \subset T \times T \times N$ and 
		$Y \times_{M^2} T^2 \subset T \times T \times N'$, respectively, on the first factor $T$.
		\item $\eta_Y: T \times T \times N \to T \times T \times N \times T \times
		N'$ is the map given by 
		\[
		\eta_Y=p_1\times P_1\circ \gamma_y \circ p_2 \times P_2\circ \gamma_y \circ p_2 \times p_3 \times P_3\circ \gamma_y \circ p_2
		\]
		
	\end{itemize}
  
  It follow immediately from these definitions that:
  $$
  p_2 \gamma_X = f \qquad p_{123} \circ \eta_Y = \id  
  $$
  and that $p^*_{145} I_Y \cong \eta_{Y*} \Oo_{T \times T \times N}$, and $I_X \cong \gamma_{X*} \Oo_T$.

  It follows from the definition of relative sum and
  Lemma~\ref{relslope-proj} that
  $$
  p_{16*}(\p_1 \times F_n \circ p_{24})(p_{123}^*I_X \otimes p_{145}^* I_Y)
  $$
  is the ideal that cuts out the graph of
  $$
  \tau_n(X + Y/G^2 \times N
  \times N',  (P_1,P_2,P_3,P_4,P_5,P_2+P_4)
  $$
  in $T \times T$. 

  Then applying Lemma~\ref{proj-formula} and simplifying, we obtain
  $$
  \begin{array}{cl}
    & p_{16*}(p_1 \times F_n \circ p_{24})_*(p_{123}^* I_X \otimes
      p_{145}^* I_Y) = \\
  = & p_{16*}(p_1 \times F_n \circ p_{24})_*(p_{123}^* I_X \otimes
      \eta_{Y*} \Oo_{T \times T \times N}) =  \\
  = & p_{16*}(p_1 \times F_n \circ p_{24})_*\eta_{Y*}(\eta_Y^*p_{123}^* I_X \otimes
       \Oo_{T \times T \times N}) =  \\
  = & p_{16*}(p_1 \times F_n \circ
      p_{24})_*\eta_{Y*} \eta_Y^*p_{123}^* \gamma_{X*} \Oo_T = \\
  = & (p_{16} \circ (p_1 \times F_n \circ
      p_{24}) \circ \eta_{Y} \circ \gamma_{X}) \Oo_T.\\
  \end{array}
  $$

  The composition
  $p_{16} \circ (p_1 \times F_n \circ p_{24}) \circ \eta_{Y} \circ
  \gamma_{X}: T \to T$ is the morpmism
  $ p_1 \times F_n \circ f\times g$: indeed, the diagram
  $$
  \xymatrix{
    T \ar[r]^-{\gamma_X} \ar[dr]_{f \times g} & T \times T \times N
    \ar[r]^-{\eta_Y} & T
    \times T \times N \times T \times N' \ar[d]^{p_1 \times F_n \circ p_{24}} \\
    & T \times T \ar[dr]_{F_n} & \ar[l]^-{p_{24}} T
    \times T \times N \times T \times N' \times T \ar[d]^{p_6}  \\
    & & T \\
  }
  $$
  commutes, the commutation of the  upper cycle follows from the
  definitions of $\gamma_X$ and $\eta_Y$, and commutation of the lower
  cycle is immediate.

  The conclusion of the proposition follows.
\end{proof}

An element of $\End(k[\eps]/(\eps^{n+1}))$ is completely determined by
the truncated polynomial with zero constant term which is the value of
the endomorphism on the generator $\eps$; addition of such truncated
polynomials defines addition operation on $\End(k[\eps]/(\eps^{n+1}))$
with respect to which composition of endomorphisms is distributive,
turning $\End(k[\eps]/(\eps^{n+1}))$ into a ring.

\begin{cor}
  \label{end-p1-is-k} With the ring structure described above,
  $\End(k[\eps]/(\eps^2))$ is isomorphic to $k$.
\end{cor}

\begin{proof}
  This is a consequence of Propositions~\ref{rel-composition} and
  \ref{addition}.
\end{proof}


\section{Differential equations in formal power series}
\label{equations}

\subsection{Uniqueness of solutions}

We will be using the uniqueness of solutions of differential equations
in formal power series over a field of characteristic $0$ on several
occasions. Though it is well-known we include the short proof for the
sake of completeness. It is stated precisely in the form needed for
us.

\begin{lemma}[Picard-Lindel\"of for formal power series]
  \label{ode-unique} Let $k$ be a field of characteristic 0.  Let
  $f(x,y) \in k[[x,y]]$ be a formal series. Then there exists a unique
  $y \in xk[[x]]$ such that
  $$
  y' = f(x,y)
  $$
  where $y'$ is a formal derivative.
\end{lemma}

\begin{proof}
  Denote $f(x,y):=\sum\limits_{i,j=0}^{\infty}f_{i,j}x^{i}y^j$ and
  define a map $ \varphi: xk[[x]] \to xk[[x]]$ by setting
  $$
  y \mapsto \int_0^x f(\alpha,y) d\alpha := \sum_{i,j=0}^\infty
  f_{i,j} \int_0^x \alpha^i y^j d\alpha
  $$
  where $\int_0^x$ denotes formal antiderivative with zero constant
  term. Then $\varphi$ is well defined, and we claim that it is a
  contracting map. Indeed, let $v: k[[x]] \to \Z$ be the valuation on
  the ring of formal series, $\norm{\cdot}_v$ the associated norm
  $\norm{x}_v=e^{-v(x)}$ (which is trivially 1 on constants).
  $$
  \norm{\varphi(y_1) - \varphi(y_0)}_v= \norm{ \sum_{i,j=0}^\infty
    f_{i,j} \int_0^x \alpha^i (y_1^j - y_0^j)}_v d\alpha \leq
  \dfrac{1}{e} \norm{y_1 - y_0}_v
  $$
  As $(k[[x]], ||\cdot||_v)$ is a complete metric space, the Banach
  fixed point theorem asserts that $\varphi$ has a unique fixed
  point. By definition of $\phi$ this fixed point is a solution of the
  given differential equation.
\end{proof}

\subsection{Differential equations for slopes}
\label{c-systems}

On $\A^1$ there are natural local coordinate systems at any point
$x_0$ given by the projective system of morphisms
$k[x] \to k[\eps]/(\eps^n)$, $x \mapsto \eps + x_0$. We would like to
have in a similar vein a coherent choice of local coordinate systems
at every point of a dense open subset of an arbitrary curve. Let $U$
be a curve and suppose an \'etale morphism $u: U\to \A^1$ is chosen,
then for any closed point $y \in U$ the pullback of
$x - u(y) \in k[x]$ gives a canonical choice of uniformizer in
$\Oo_{U,y}$, and hence local coordinate systems are chosen at each
such $y$; these are the local coordinate systems coming from the 
isomorphisms $\widehat{\Oo_{U,y}} \cong\widehat{\Oo_{\A^1, u(y)}}$. We
will refer to such local coordinate systems as \emph{liftings} of
local coordinate systems on $\A^1$.

\begin{lemma}
  \label{coord-system} Let $u, v$ be two \'etale maps from $U, V$ to
  $\A^1$, and let $p$ be the product map
  $p:=u \times v: U \times V \to \A^2$. Then for any $n > 0$, for any
  closed subscheme $Z$ of $U\times V$ and any point $Q \in Z$
  $$
  \tau_n(Z, Q) = \tau_n(p(Z), p(Q))
  $$
  where the local coordinate systems on $U \times V$ and on $\A^2$ are
  chosen as above.
\end{lemma}

\begin{proof}
  By Lemma~\ref{slope-graph},
  $S := \Spec \Oo_{Z,Q} \otimes_{\Oo_{U \times V, Q}} k[\eps,
    \eta]/(\eps^n, \eta^n)$ is the graph of $\tau_n(Z,Q)$, and
    $\Spec \Oo_{p(Z),p(Q)} \otimes_{\Oo_{\A^2, p(Q)} k[\eps,
      \eta]/(\eps^n, \eta^n)} = p(S)$ is the graph of
      $\tau_n(p(Z), p(Q))$. It is left to notice that the map
      $u \times v$ induces an isomorphism between
      $\Spec \Oo_{U \times V, Q} \otimes k[\eps, \eta]/(\eps^n,
      \eta^n)$ and
      $\Spec \Oo_{\A^2, p(Q)} \otimes k[\eps, \eta]/(\eps^n, \eta^n)$,
      by \'etaleness and the choice of local coordinate systems, and
      furthermore, $S$ and $p(S)$ are isomorphic.
\end{proof}

\begin{lemma}
  \label{expansion-eq} Let $Z \subset \A^2$ be a curve defined by an
  equation $h(x,y)=0$. Let $Q \in Z$ be a regular point such that the
  projection of $Z$ on the first factor is \'etale. Let $f \in k[[x]]$
  be the formal power series expansion of $Z$ in the natural local
  coordinate system on $\A^2$ at $Q$. Then $h(x,f)=0$.
\end{lemma}

\begin{proof}
  Without loss of generality one may assume $Q=(0,0)$. Then the fact
  that $n$-th order slope of $Z$ is the endomorphism $f_n$ of
  $k[\eps]/(\eps^n)$ means that the morphism of algebras
  $$
  k[x,y] \to k[\eps]/(\eps^n)\qquad x \mapsto \eps, \: y \mapsto
  f_n(\eps)
  $$
 factors through $k[x,y]/h$, i.e. $h(x,f_n)=0 \mod x^n$ [It may be
 worth giving slightly more detail why this is true. Maybe as an
 example following the definition of slope. This is also used below,
 so would be good to have explicitly, and will also clarify the
 definition of slope]. The
  conclusion of the lemma follows by passing to the limit.
\end{proof}

Let $U,V$ be open subsets of $M$, and let $u: U \to \A^1, v: V \to
\A^1$ be \'etale maps.  Let $Z \subset U \times V$ be a regular curve
such that the projection of $Z$ on $U$ is \'etale, and for any point
$Q=(Q_1, Q_2) \in U \times V$ the first order slope of $Z$ at $Q$ is
well-defined (with respect to the liftings of the natural local
coordinate systems at $u(Q_1), v(Q_2)$ to $Q_1, Q_2$) provided by the
Lemma \ref{coord-system}). Let $Z' \subset U \times V \times \A^1$ be
the closed curve such that $(x, y, s) \in Z'$ if $s$ is the first
order slope of $Z$ at $(x, y)$.

\begin{prop}
  \label{slope} In the above setting, Let $k[[u,v]] \cong \widehat{
  \Oo_{M^2,Q}}$ be the isomorphism given by the chosen local
  coordinate system at $Q$, and $\eta: \Spec k[[x]] \to Z$ the
  morphism given by the morphism of rings
  $$
  \eta^*: \Oo_{Z,Q} \to k[[x]], u \mapsto x, v \mapsto f
  $$
  where $f\in xk[[x]]$. Consider the morphism $\eta': \Spec k[[x]] \to
  U \times V \times \A^1$ defined by the morphism of rings
  $$
  \eta'^*: \Oo_{U \times V \times \A^1, Q_1 \times \{0\}} \to k[[x]], \:
  u \mapsto x, \:v \mapsto f, \: w \mapsto f'
  $$
  where $f'$ is the formal derivative of $f$. Then $\eta'$
  factors through $Z'$.
\end{prop}
 
\begin{proof}
  By Lemma~\ref{coord-system} the slope of $Z$ at $Q$ is the same as
  the slope of $(u \times v)(Z)$ at $(u(Q_1), v(Q_2))$, in their
  respective local coordinate systems. So it suffices to prove the
  statement assuming that $Z$ is a locally closed subset of
  $\A^2$. Without loss of generality we may assume $Q=(0,0)$.
  
  Let $Z$ be defined by a polynomial $h(u,v)$. The slope of $Z$ at a
  point $(x,y) \in \A^2$ is given by the expression
  $$
  \dfrac{\dfrac{\partial
      h}{\partial u}(x,y)}{\dfrac{\partial h}{\partial v}(x, y)}
  $$
  where the derivatives are formal.  Let $f \in k[[x]]$ be a formal
  power series such that $h(x,f)=0$ as provided by Lemma
  \ref{expansion-eq}. Then
  $$
  \dfrac{d}{dx}h(x,f) = \dfrac{\partial h}{\partial u}(x,f) +
  \dfrac{\partial h}{\partial v}(x,f)f'
  $$
  by the chain rule (Corollaire~1, \citep{algebre}, IV.\S 6), and
  $\dfrac{d}{dx} h(x,f)=0$ by our choice of $f$. Therefore
  $$
  \dfrac{\dfrac{\partial h}{\partial
    u}(x,f)}{\dfrac{\partial h}{\partial v}(x, f)} = f'
  $$  
  and $(x,f,f') \in Z'$.
\end{proof}

\begin{cor}
  \label{distrib-ode} Let $u: U \to \A^1$, $v: V \to \A^1$ be \'etale,
  $Z \subset U \times V \times \A^1$ be a closed subset that is
  \'etale over $U \times V$ and $Q\in U\times V$. Given a curve $X$
  incident to $Q$ whose projection on $U$ is \'etale let
  $X' \subset X \times \A^1$ be a 1-dimensional closed subset such
  that $(x,y,s) \in X'$ if the first order slope of $X$ at the point
  $(x,y)$ (in the local coordinate system lifted from the natural
  local coordinate system at $(u(x), v(x))$) is $s$.

  Then there exist a formal power series $h \in k[[x,y]]$ such that for
  any $X$ as above, with a fixed slope at $Q$, and such that
  $X' \subset Z$, letting 
  $f = \varprojlim\limits_n \tau_n(X, Q)(x) \in k[[x]]$ be the formal
  power series expansion of $X$ at $Q$, we have:
  $$
  f' = h(x,f)
  $$
  where $f'$ is the formal derivative of $f$.
\end{cor}

In the statement of this corollary $Z$ should be regarded a
``multi-valued distribution'', a variety that specified what slopes
$X$ is allowed to have at a particular point.

\begin{proof}
  By the same reasoning as in the proof of Lemma~\ref{slope} we can
  reduce the situation to $U, V$ open subsets of $\A^1$ and $X$ a
  locally closed subset of $\A^2$.

  Let $s_0 = \tau_1(X,0)$, we have the following commutative diagram
  $$
  \xymatrix{
    \widehat{\Oo_{\A^3, (0,0,s_0)}} \cong k[[x,y,z]]  \ar[r]^-{f_1} &
    \widehat{\Oo_{Z,0}} \ar[dl]_\sim^g \ar[r]^-{f_2}  & \widehat{\Oo_{X',(0,0,s_0)}} \\
    \widehat{\Oo_{\A^2, (0,0)}} \cong k[[x,y]] \ar[rr] \ar[u] &  &
    \widehat{\Oo_{X,(0,0)}} \ar[u] \\
  }
  $$
  where $g$ is an isomorphism by \'etaleness of the projection of $Z$
  on $\A^2$ and (identifying the local rings $\widehat{\Oo_{}}$ via $g$) the map $f_1$ is given on generators by
  \[
  x \mapsto x,\:  y \mapsto y, \: z \mapsto h
  \] for some $h \in k[[x,y]]$.
  On the other hand, by Proposition~\ref{slope}, the compostion $f_2\circ f_2$ is given on generators by 
  \[x \mapsto x, \: y \mapsto f, \: z \mapsto f'\] with the desired conclusion.
\end{proof}

\subsection{Divided power structures and an ODE for $\G_m$}

In Section~5 we will be looking at curves is the product of two
copies of the multiplicative group $\G_m \times \G_m$, and in
connection with this we will need to consider formal power series
solutions to the differential equation
\begin{equation}
y'=a \cdot \dfrac{1+y}{1+x}
\label{ode-mult}
\end{equation}
Below we collect some facts about solutions to this equation,
the situation in positive characteristic requiring a somewhat subtle treatment.

Let $k$ be a field of characteristic 0. Consider \emph{binomial power
  series} for $a \in k$, defined as
$$
(1+x)^a = \sum_{k=0}^\infty \dfrac{a \cdot (a-1) \ldots (a-k)}{k!} x^k
$$
If $a$ is integer then this is a polynomial, but the formal power
series are well-defined for any $a \in k$.

\begin{prop}
  \label{solutions-slope-gm-char0} Let $k$ be a field of
  characteristic $0$. The unique solution of the differential equation
  $(\ref{ode-mult})$ is the binomial power series $(1+x)^a -
  1$. These formal power series are algebraic over $k[x]$ only if
  $a$ is rational.
\end{prop}

\begin{proof}
  Observe that $\dfrac{d}{dx} (1 + x)^a =  a(1+x)^{a-1}$, it is
  easy to check that $(1+x)^a$ is a solution by direct substitution. 

  For the second point observe that if $a$ is not rational, then
  powers of $(1 + x)^a$ generate a module of infinite rank over
  $k[x]$, and so $(1 + x)^a$ is not algebraic. One uses the property
  $$
  (1 + x)^a(1 + x)^b=(1 + x)^{a+b}
  $$
  which can be easily derived even for $a$ not rational (see Section~2
  of \citep{goss2015polynomials} and references therein).
\end{proof}

Though uniqueness of solutions of ODEs fails in positive
characteristic, in general, something can still be said concerning
solutions of the equation $(\ref{ode-mult})$. Unfortunately, the
binomial power series are not even well-defined in positive
characteristic, because integers dividing the characteristic appear in
denominators of the coefficients.

In order to remedy that, we consider solutions of differential
equation $(\ref{ode-mult})$ in a bigger ring into which $k[[x]]$ can
be mapped. This ring is the completion of the divided power
polynomials ring $k\la x \ra$ of $k[x]$ with respect to a certain
filtration. For definition and properties of divided power structures
we refer to \citep{berthelot1974cohomologie}, see also
\citep{baker1995differential}.

The facts about $k\la x \ra$ that we will need are few. The ring 
$k\la x \ra$ is generated over $k$ by variables $x^{[n]}, n \in \N$,
subject to the relations
$$
x^{[n]} x^{[m]} = \dfrac{(n+m)!}{n!m!} x^{[n+m]}
$$
($x^{[0]}$ customarily means 1).

Consider the homomorphism $\varphi: k[x] \to k\la x \ra$ sending
$x$ to $x^{[1]}$. The kernel of this morphism is the ideal generated
by $x^p$. Define the derivation $D_x$ on the generators by
$D_x x^{[n]} = x^{[n-1]}$. The homomorphism $\varphi$ has the
following property:
$$
\varphi(p'(x)) = D_x \varphi(p(x))
$$

Consider the completion $\widehat{k\la x \ra}$ of $k\la x \ra$ with
respect to the filtration by the ideals generated by sequences of
elements of the form $x, x^{[2]}, x^{[3]}, \ldots, x^{[i]}$. The
homomorphism $\varphi$ above extends to a morphism
$\wbar{\varphi}: k[[x]] \to \widehat{k\la x \ra}$, and the derivation
$D_x$ is extended in a unique way to $\widehat{k\la x \ra}$. The
compatibility of derivations is preserved:
$\wbar\varphi(p'(x)) = D_x \wbar\varphi(p(x))$.

Define \emph{divided power binomial series}
$$
(1 + x)^a = \sum_{n=0}^\infty a \cdot (a-1) \cdot \ldots \cdot (a-n+1) x^{[n]}
$$
in $\widehat{k\la x \ra}$. 

\begin{lemma}
  \label{solutions-slope-gm-charp} Let $k$ be a field of positive
  characteristic $p > 0$. The differential equation $(\ref{ode-mult})$
  has a solution
  $$
  (1+x)^a - 1
  $$
  in  $\widehat{k\la x \ra}$. It is in $k[[x]]$ if and only if $a \in \F_p$. Let $y_1, y_2 \in k[[x]]$ be two distinct solutions, then
  $\dfrac{y_1 + 1}{y_2 + 1}$ belongs to $k((x^p))$ and is
  non-constant.
\end{lemma}

\begin{proof}
  Observe that if $f\in k[[x]]$ is a solution to $(\ref{ode-mult})$,
  then $\wbar\varphi(f)$ is a solution to $(\ref{ode-mult})$ (with
  derivation interpreted as $D_x$). 

  If $a \in \F_p$, the binomial formal power series is a polynomial
  which is a solution to equation $\ref{ode-mult}$ by direct
  verification. If $a \notin \F_p$ then the divided powers binomial
  power series is a solution to $(\ref{ode-mult})$ by direct
  verification.  Further, one checks then that
  $(1+x)^a \in \widehat{k\la x\ra}$ is not in the image of $\varphi$
  (since it has $x^{[n]}$ terms for $n \geq p$) and so there are no
  solutions in $k[[x]]$.


  Let $y_1$, $y_2$ be distinct solutions in $\widehat{k\la x \ra}$ of the
  equation $(\ref{ode-mult})$. Then
  $$
  D_x \left(\dfrac{y_1 + 1}{y_2 +1}\right) = \dfrac{y_1'(y_2+1) -
    y_2'(y+1)}{(y_2+1)^2} = \dfrac{a(y_1+1)(y_2+1) -
    a(y_2+1)(y_1+1)}{(1+x)(y_2+1)^2} = 0
  $$
  and therefore $\dfrac{y_1 + 1}{y_2 +1}$ is a constant. One checks
  that if $\dfrac{y_1 + 1}{y_2 +1}$ is a constant then it must be
  1. The same argument in $k[[x]]$ yields that formal power series
  solutions $y_1, y_2$ have the property that $\dfrac{y_1 + 1}{y_2 +
    1} \in k((x^p))$.

  % If $a \in \F_p$ then an algebraic solution $y_1$ must have the
  % property that $\dfrac{y_1 + 1}{(1+x)^a}$ is algebraic.

\end{proof}



\section{Interpretation of a field from a pure-dimensional witness
  family}
\label{interp-field}

Using the machinery developed in the last two sections, we will now
start our construction of the field configuration. As explained in the
introduction, we will first construct a group configuration, and then
exploit it to construct the field configuration.


\subsection{Notation, conventions and some model theory}\label{sec:parameters}
We remind that $M$ is an algebraic curve over an algebraically closed
field, which we may assume to be regular. We fix some
$X\subseteq M^2\times T$, the total space of an ample (almost)
faithful family of curves in $M^2$. We let $\mathfrak M$ denote the
full Zarski structure on $M$ and $\Mm$ the reduct $(M,X)$.
%Restricting to the regular locus of $M$ we may assume that $M$ is
%regular, which will be the standing assumption in the present section.
%
%A reduct of $M$ is non-locally modular if and only if there exists a
%\emph{faithful} (in the terminology of \citep{zg}; the term
%\emph{normal} is commonly used in the literature) definable family of
%strongly minimal sets $X \subset M^2 \times T$, i.e. a family such
%that $\dim T = 2$, $\dim X_t=1$ for $t \in U$, $\dim T \wo U \leq 1$
%and such that $\dim (X_t \wo X_s) \cup (X_s \wo X_t) = 0$ for
%$t \neq s$.  Let $M$ be a one-dimensional algebraic curve. Throughout
%this section the family of curves that witness of lack of local
%modularity of $\Mm$ will be denoted $X$ with parameter space $T$,
%unless this notation is explicitly claimed to be used for something
%else. 
%
We fix the following conventions:

\begin{defn} Let $M$ be as above. 
\begin{enumerate}
\item By a \emph{family of curves} in $S$ (some variety $S$) we
  understand a locally closed subset $X\subseteq S\times T$ for some
  $T$ all of whose fibres, $X_t$, are 1-dimensional (not necessarily
  irreducible);
\item by a \emph{family of curves} in $M^2 \times N$ for some variety
  $N$ we understand a family of curves in $M^2\times N$ whose
  projection onto $M^2$ has finite fibres.
%	a locally closed subset
%	$X\subseteq M^2 \times N \times T$ for some $T$ such that the fibres
%	(which are not assumed irreducible) $X_t$ are of dimension 1 for all
%	$t\in T$ and such that projection on $M^2$ has finite fibres;
\item a family of curves is \emph{pure-dimensional} if all irreducible
  components of the fibres, $X_t$, are of the same dimension for all
  $t\in U\subseteq T$ with $U$ dense and open;
\item the family of \emph{(scheme-theoretic) intersections} of two
  families of curves $X\subseteq M^2\times T$ and
  $Y\subseteq M^2\times S$ is the surjective morphism
  $(X \times S) \times_{M^2 \times T \times S} (Y \times T) \to T
  \times S$ (that is, the family of closed subschemes of $M^2$ whose
  fibre over $(t,s)$ is $X_t\times_{M^2} Y_s$);
\item the family of \emph{(scheme-theoretic) relative intersections}
  of two families of curves $X\subseteq M^2 \times N \times T$ and
  $Y\subseteq M^2 \times N' \times S$ is the surjective morphism
  \[
    (X \times S) \times_{M^2 \times T \times S} (Y \times T) \to T  \times S
  \] 
  (that is, the family of closed subschemes of
  $M^2 \times N \times N'$ whose fibre over $(t,s)$ is
  $X_t\times_{M^2} Y_s$). This coincides with the notion for families
  of curves in $M^2$ defined above if one takes $N$ and $N'$ to be
  points.
\end{enumerate}
\end{defn}

\textbf{Throughout this section we assume that $X$ is a family of pure
  dimensional curves.} This significant reduction will be justified in
Section \ref{pure-dimensional}.

To state and prove the results of the present section (and to an even
greater extent the next section) we fix some notation, as well as some
standard model theoretic terminology.

The notion of generic points\footnote{This model theoretic terminology
  refers to $K$-rational points of a variety, and differs from the
  standard algebro-geometric notion of the same name.} plays an
important role in all that follows. Recall from Section
\ref{background} that the full Zariski structure $\mathfrak M$ of an
algebraic curve $M$ over $k$ extends naturally to any algebraically
closed $K\ge k$. Given a Zariski closed set $S$ over $k$ and a point
$p\in S(K)$ we say that $p$ is $\mathfrak M$-\emph{generic} (over $k$)
if the locus of $p$ over $k$ is $S$. While we cannot, a priori, define
the $\Mm$-locus of a point, we can define its rank, denoted
$\mathrm{MR}$, and if $S$ is $\Mm$-definable (not necessarily Zariski
closed) over $k$ we say that $p\in S(K)$ is $\Mm$-generic (over $k$)
if $\mathrm{MR}(S')\ge \mathrm{MR}(S)$ for any $S'$ $\Mm$-definable
over $k$ such that $p\in S'$.

The $\Mm$-type of a point $p\in M^n(K)$ over $k$, denoted
$\tp_{\Mm}(p/k)$ is the collection of all $\Mm$-definable sets $S$
such that $p\in S$. We set
$\mathrm{\tp_\Mm(p/k)}=\min\{\mathrm{MR}(S): S\in \tp_\Mm(p/k)\}$. In
those terms, $p$ is $\Mm$-generic in $S$ if
$\mathrm{MR}(\tp_{\Mm}(p/k))=\mathrm{MR}(S)$. In a similar way we
define $\tp_{\mathfrak M}(p/k)$. We note that while the Morley rank of
an $\Mm$-definable set is the same, computed in $\Mm$ and in
$\mathfrak M$, the same is not true for the Morley rank of the type of
a point. For that reason we have to distinguish $\Mm$-generic points
from $\mathfrak M$-generic points. This gives rise to respective
notions of independence. Namely, $p$ is independent from $A$ over $k$
if $\mathrm{MR}(\tp(p/k))=\mathrm{MR}(\tp(p/\acl(kA)))$. This can be
used to define the notion of a canonical parameter of a set. Say that
$s$ is a \emph{canonical parameter}\footnote{The definition given here
  is slightly weaker than the standard one, but it suffice for our
  needs in this paper.} for the definable set $S$ if whenever $p\in S$
is generic $p$ is independent from $s$. We do not claim that $s$ is
uniquely determined, but $\mathrm{MR}(\tp(s/k))$ depends only on $S$
(not on $s$). It is useful to note that if $X\to T$ is a faithful
family of plane curves then $t$ is a canonical parameter for $X_t$.

A point $p$ is \emph{algebraic} over $k$ if
$\mathrm{MR}(\tp(p/k))=0$. We let $\acl_\Mm(A)$ be the set of
$\Mm$-algebraic points over a set $A$, and similarly
$\acl_\mathfrak{M}(A)$ (this coincides with algebraic closure in the
field theoretic sense). The observations of the previous paragraph
imply that $\acl_\mathfrak{M}{(A)}$ is -- a priori -- finer than
$\acl_\Mm(A)$. Most of Section \ref{pure-dimensional} can be viewed as
a proof that, up to naming constants, the two notions coincide.


Given a family $Y \subset M^2 \times S$ of curves and a point
$Q\in M^2$ we let $Y^Q \to S^Q$ denote the subfamily of curves
incident to the point $Q$. The above shows that
$\mathrm{MR}(Y^Q)=\mathrm{MR}(Y)-1$ for $\Mm$-generic $Q$. Moreover,
if $Y$ is $\Mm$-stationary so is $Y^Q$ (for $Q$ generic in $\Mm$). We
will be using these observations implicitly throughout the
text. Similarly, given a family of curves
$Y \subset M^2 \times N \times S$ we denote $Y^Q \to S^Q$ the
subfamily of curves whose projection on $M^2$ contains $Q$. We will
call such curves \emph{relatively incident to $Q$}.

One of our main tools for manipulating curves is their
composition. For a family of curves $X$, and a curve
$Z \subset M^2 \times N$ we let $Z \circ X \to T$ be the family whose
fibres are $(Z \circ X)_t = Z \circ X_t$; similarly, $X \circ Z$
denotes the family $(X \circ Z)_t = X_t \circ Z$. Note that if $X$ is
faitful then so is $Z\circ X$, implying that
$\mathrm{MR}(Z\circ X)=\mathrm{MR}(X)$.

More generally, given two (definable) families
$Y_1 \subset S_1 \times M^2$, $Y_2 \subset S_2 \times M^2$ of curves
in $M^2$ we denote $Y_1 \circ Y_2$ the (definable) family parametrized
by $S_1 \times S_2$ with the curve $(Y_1)_{s_1} \circ (Y_2)_{s_2}$
corresponding to the parameters $s_1, s_2$. Note, however, that even
if $Y_1$, $Y_2$ are faithful the composition family $Y_1\circ Y_2$
need not be faithful, in which case we let $Y_1\circ Y_2$ denote the
\emph{normalisation} of $Y_1\circ Y_2$. It is useful to observe that,
by the previous paragraph, even after this normalisation
$\mathrm{MR}(Y_1\circ Y_2)\ge \max\{\mathrm{MR}(Y_1),
\mathrm{MR}(Y_2)\}$.

\subsection{The group and field configurations}\label{configurations}
In the strongly minimal context, certain combinatorial configurations
of (imaginary) elements are known to exist only in the presence of a
definable group or a definable field. It is by constructing such
configurations that the main theorem  of the present paper is
proved. We will now describe these configurations in more detail: 

\begin{defn}[Group configuration]
	Let $M$ be a model of a strongly minimal theory, and let $\dim$ the
	be associated dimension function on tuples.
	$$
	\groupconf{a}{b}{c}{x}{y}{z}
	$$
	%   \xy <1.5cm,0cm>: (1,0.2)*{x}; (2,0.2)*{y}; (0.8,-0.8)*{z};
	%   (-0.2,0)*{a}; (-0.2,-1)*{b}; (-0.2,-2)*{c}; (1,0)*{}; (0,-2)*{}
	%   **\dir{-}; (0,-2)*{}; (0,0)*{} **\dir{-}; (0,0)*{}; (2,0)*{}
	%   **\dir{-}; (2,0)*{}; (0,-1)*{} **\dir{-};
	%   \endxy
	
	The set $\set{a,b,c,x,y,z}$ of tuples is called a \emph{group
		configuration} if there exists an integer $n$ such that
	\begin{itemize}
		\item all elements of the diagram are pairwise independent and
		$\dim(a,b,c,x,y,z)=2n+1$;
		\item $\dim a = \dim b = \dim c=n$, $\dim x = \dim y = \dim z = 1$;
		\item all triples of tuples lying on the same line are
                  dependent, 	and moreover, $\dim(a,b,c)=2n$, 
		$\dim(a,x,y)=\dim(b,z,y)=\dim(c,x,z)=n+1$;
	\end{itemize}
\end{defn}

If $G$ is a connected group definable in a strongly minimal theory,
acting transitively on a strongly minimal definable set $X$, then one
can construct a group configuration as follows: let $g,h$ be
independent realisations of the generic type of $G$ and let $x$ be a
realisation of a generic type of $X$, then $\tuple{g,h,g\cdot h, u, g
	\cdot u, g \cdot h \cdot u}$ is a group configuration.

\begin{fact}[Hrushovski]
	\label{hrugrou} Let $M$ be a strongly minimal structure and let
	$\tuple{a,b,c,x,y,z}$ be a group configuration. Then there exists a
	definable group $G$ acting transitively on a strongly minimal set
	$X$ with the associated group configuration $\tuple{g,h,g\cdot h, u,
		g \cdot
		u, g \cdot h \cdot u}$ such that $\acl(a)=\acl(g)$, 
	$\acl(b)=\acl(h)$, $\acl(g\cdot h)=\acl(c)$, $\acl(x)=\acl(u)$,
	$\acl(y)=\acl(g\cdot u)$, $\acl(z)=\acl(g \cdot h \cdot u)$. In
	particular, $\dim G=\dim a$.
\end{fact}

This follows from the main theorem of \citep{bouscaren-group} and the
fact that infinitely definable groups in stable theories are
intersections of definable groups (see, for example,
Theorem~5.18\citep{poizat}). The original proofs of these statements
are contained in \citep{hru-phd}.

\begin{fact}
	\label{hrugrou-faithfulness} If in the statement of
	Fact~\ref{hrugrou} one requires that the canonical base of
	$\tp(x,y/a)$ is interalgebraic with $a$ and similarly for
	$\tp(z,y/b)$ and $\tp(z,x/c)$ then the action of $G$ on $X$ is
	faithful.
\end{fact}

\begin{fact}[Hrushovski]
	\label{fieldconf} Let $G$ be a group of Morley rank $n>1$ acting
	transitively and faithfully on a strongly minimal set $X$. Then
	there exists a definable field structure on $X$ and either $n=2$ and
	$G \cong \G_a \rtimes \G_m$, or $n=3$ and $G=\mathrm{PSL}_2$.
\end{fact}

The original reference is \citep{hru-phd}, an exposition can also be
found in \citep{poizat} (Theorem 3.27). Establishing that $G$
isomorphic to $\G_a \rtimes \G_m$ or to $\mathrm{PSL}_2$ is the
crucial point in the proof of Fact~\ref{fieldconf}. In the present
context, where $G$ and $X$ are definable in an algebraically closed
field (rather, the full Zariski structure on an algebraic curve) this
statement can be proved directly and much more easily.


\subsection{Generically \'etale projections}

In order for slopes (defined in Section~\ref{slopes}) to be
well-defined we need projections of curves in $M \times M$ on the
first factor $M$ to be \'etale in a Zariski open neighbourhood of the
point. The following lemma asserts that at least one of the
``coordinate'' projections of a curve $X \subset M \times M$ does
satisfy this requirement generically. The statement of the lemma is
obvious in characteristic 0, but the existence of everywhere ramified
morphisms in positive characteristic makes it non-trivial in this
case.

\begin{lemma}
  \label{projections-etale} Let $M$ be a one-dimensional closed
  irreducible curve over a field of positive characteristic. Let
  $X \subset M \times M$ be an irreducible closed curve, 
  $p_1, p_2: X \to M$ the projections on the respective factors $M$.
  Then there exists a dense open $O \subset M$ such that either $p_1$
  restricted to $X \cap O \times M$ or $p_2$ restricted to
  $X \cap M \times O$ is \'etale.
\end{lemma}

\begin{proof} The lemma is clear if one of the projections $p_i$ is
not dominant. So we may assume that this is not the case.

Let $\Omega_{M/k}$, $\Omega_{X/k}$ be the sheaves of modules of
K\"ahler differentials over $k$ of $M$ and of $X$
respectively. Consider the natural map of sheaves
$$
p_1^* \Omega_{M/k} \oplus p_2^* \Omega_{M/k} \to \Omega_{X/k}
$$
which is surjective because $X$ is embedded into $M^2$. Localising at
the generic point $\chi$ of $X$ we get a surjective map, $f$, of
vector spaces over the field $k(\chi) = k(X)$
  $$
  f: p_1^* \Omega_{M/k} \otimes k(\chi) \oplus p_2^* \Omega_{M/k} \otimes
  k(\chi) \to \Omega_{X/k} \otimes k(\chi).
  $$
  Then $f=f_1\oplus f_2$ where
  $f_i: p_i^* \Omega_{M/k} \otimes k(\chi) \to \Omega_X \otimes
  k(\chi)$ is a $k(X)$-linear map.  As $f$ is surjective and
  $\dim_{k(X)}\Omega_X \otimes k(\chi)=1$, at least one function out
  of $f_1$ and $f_2$ must be an isomorphism.
  
  Assume that for example $f_1$ is an isomorphism. Then the sheaf
  $\Omega_{X/M}$ of relative differentials on $X$ over $M$ with
  respect to the first projection, is isomorphic over a dense open set
  to the structure sheaf. So $p_1$ is generically unramified. The
  statement of the lemma then follows from generic flatness
  (Fact~\ref{generic-flatness}).
\end{proof}

Recall that if $f: X \to Y$ is a morphism of schemes over a field of
characteristic $p$ then $\Fr_{X/Y}: X \to X^{(p)}$, the \emph{relative
  Frobenius morphism} is defined to be $\Fr_X \times \id_Y$ where
$\Fr_X$ is the absolute Frobenius morphism (the morphism defined by
raising the functions in the structure sheaf to the $p$-th power), and
$\id_Y$ is the identity on $Y$.

\begin{lemma}
  \label{cancel-insep}
  Assume that the field of definition of $M$ is of positive
  charactrestic $p$, and assume that the projection of $X$ on the
  first $M$ is \'etale.  Then there exists a number $n$ such that both
  projections, on $M$ and on $M^{(p^n)}$, of the preimage $X'$ of
  curve $X$ under the map
  $\Fr_{M^2/M}^n: M \times M^{1/p^n} \to M \times M$, where
  $\Fr_{M^2/M}$ is the relative Frobenius morphism, are generically
  \'etale.
\end{lemma}

\begin{proof}
  Without loss of generality assume that $f_1$ is an
  isomorphism. Therefore $f_2=0$ which implies that $p_2$ induces an
  extension of function fields $k(M) \subset k(X)$ that factors into a
  sequence of a separable extension an a purely inseparable one:
  $k(M) \subset k(X') \subset k(X)$. By definition of the relative
  Frobenius, it raises the elements of $k(X)$ that do not belong to
  $k(M)$ to the $p$-th power, so $k(\Fr_{M^2/M}^n(X)) \subseteq k(X')$
  for $n$ equal to the degree of inseparability of the extension
  $[k(X'):k(X)]$.
\end{proof}

\begin{lemma}
  \label{projections-etale-family} Let $M$ be a one-dimensional closed
  irreducible curve over a field of any characteristic. Let
  $X \subset T \times M \times M$ be a family of closed curves. Then
  for any irreducible component $T_0 \subset T$, $M_0 \subset M$
  there exists a dense open $U \subseteq T_0$ and a dense open
  $O \subseteq M$ such that either $p_1$ restricted to
  $X_t \cap O \times M$ or $p_2$ restricted to $X_t \cap M \times O$
  is \'etale for all $t\in U$.
\end{lemma}

\begin{proof}
  Follows from the previous lemma applied to the generic fibre of the
  family $X \to T$.
\end{proof}

\subsection{Finding enough slopes in characteristic 0}
\label{finding-enough-slopes-0}

In order to construct a one-dimensional group configuration in $\Mm$
we need to produce families of curves (relatively) incident to a point
$Q \in M^2$ such that for some $n$ the associated $\mathfrak
M$-definable set of (relative) $n$-slopes at $Q$ almost coincides with
a one-dimensional subgroup of $\Aut(k[\eps]/(\eps^{n+1})$.  

We present two approaches. The first uses Lemma \ref{ode-unique} in a
significant way, and works only in characteristic zero. It gives a set
$O \subset M^2$, dense in an irreducible component of $M^2$, such that
$X^Q \to T^Q$ has the required propriety for any $Q \in O$. The second
approach, presented in the next sub-section, works in any
characteristic but gives a single family with the desired property for
some number $n$, in general, greater than 1.

\begin{lemma}
  \label{many-slopes} Let $M$ be an algebraic curve (not necessarily
  irreducible) over an algebraically closed field of characteristic 0,
  and let $X \subset S \times M^2$ be a $2$-dimensional faithful
  family of closed irreducible one-dimensional subsets of $M^2$. Then
  there exists an open subset $O \subset M^2$ such that for any point
  $Q \in O$ 
  \[\dim\{s\in T^Q: \tau_1(X_s,Q)\}=1.\]
\end{lemma}

\begin{proof}
  Assume not. Then there is a Zariski open set $O \subset M^2$,
  dense in an irreducible component of $M^2$, such that
  $\tau_1(X_s, Q)$ is finite for all $Q\in O$.  Pick some \'etale projections $u: U \to \A^1$, $v: V \to \A^1$. By
  Lemma~\ref{coord-system} the projections $(u \times v)(X_s)$ will
  have finitely many slopes at points of some dense open subset of
  $\A^2$, call it $O'$.
  
  Let $Z$ be the locally closed subset of $U \times V \times \A^1$
  such that a point $(x,y,s)$ belongs to $Z$ if a curve from the
  family $X$ incident to $(x,y)$ has the slope $s$ at $(x,y)$ in the
  natural local coordinate system. Shrinking $O'$ if necessary we may
  assume that $Z$ is \'etale over $O'$. Pick a point $Q \in O'$ and
  pick one of the finitely many first-order slopes the curves from the
  family $X$ has at $Q$, call it $a$. Then by
  Corollary~\ref{distrib-ode} for any curve $X_t \subset U \times V$
  with $\tau_1(X_t, Q)=a$, the formal power series expansion $f$ at a
  point $Q \in O'$ must satisfy a differential equation
  $$
  f' = h(x,f)
  $$
  for some $h \in k[[x,y]]$. By Lemma~\ref{ode-unique} there is only
  one solution $f \in xk[[x]]$ per slope value at $Q$. But according
  to our assumption about $X$ there are infinitely many curves
  incident to $Q$, which is a contradiction.
\end{proof}


\subsection{Finding enough slopes in positive characteristic}
\label{finding-enough-slopes-p}

The simplest example illustrating how the approach of the previous
section breaks down in positive characteristic is the family of curves
in $\A^2$ defined in standard coordinates $(x,y)$ by the equations
$y=x+ax^p+b$. In this family all curves have the same first order
slope 1 (with respect to the standard coordinates).  As it turns out,
the fact that the formal power series expansions of the curves differ
by a formal power series lying in $x^pk[[x^p]]$, is the only obstacle
to finding families with ``enough slopes''.

\begin{lemma}
  \label{many-slopes2} Let $M$ be a curve over an algebraically closed
  field of positive characteristic, and let $X \subset M^2 \times T$
  be an almost faithful family of curves, $\dim T = 2$.

  Then there exists a one-dimensional definable family of curves
  $Y \subset M^2\times N \times S$ incident to
  $Q' \in M^2 \times N \times S$ such that for some $n$, and some
  irreducible component $S_0 \subset S$, the set of relative slopes
  $\tau_n(Y_s/N, Q')$ as $s$ ranges in $S_0$ almost coincides with a
  one-dimensional subgroup of $\Aut(k[\eps]/(\eps^{n+1}))$.
\end{lemma}

\begin{proof}
  By Lemma~\ref{cancel-insep} for every irreducible component of $M_0$
  of $M$, for every irreducible component $T_0$ of $T$ of dimension 1
  there exists a number $n$ such that for any $t \in T_0$ generic the
  projections of
  $\Fr_{M^2/M} \times \id_T(X_t) \cap M_0 \times M_0^{(p)}$ on $M_0$
  and $M_0^{(p)}$ are generically \'etale. Fix one of such irreducible
  components $T_0$.
  
  It follows that for a generic point $Q$ in one of the irreducible
  components $M_0$, the formal power series expansions of 
  curves $X_t$ with $t$ generic in $T^Q$ are of the form $f^{p^n}$ for
  some $f \in \Aut(k[[x]])$.

  Pick a one-dimensional irreducible component $T_0 \subset T$. Then
  by Lemma~\ref{cancel-out-pth-power} for some $t_0 \in T$ the curves
  of the form $X'_t = X_t \circ X_{t_0}^{-1}, t \in T_0$, project
  separably on both factors $M$ in $M^2$. Therefore, by
  Lemma~\ref{projections-etale-family}, for a suitable point $Q'$, the
  family of curves of this form incident to $Q'$, call it $Y \to S$,
  has formal power series expansions that lie in $\Aut(k[[x]])$ for
  almost all $s \in S_0$, where $S_0$ is a one-dimensional irreducible
  component of $S$ (which is one-dimensional itself). By Krull
  Intersection theorem, there exists a smallest number $n$ such that
  the set of slopes $\tau_n(Y_s, Q')$ is one-dimensional. If $n=1$, we
  are done. Otherwise, the set of values $\tau_{n-1}(Y_s, Q')$ as $s$
  ranges in $S_0$ is finite, and actually, as $S_0$ is irreducible,
  consists of a single value. Then for some $s_1 \in S_0$, the set of
  slopes $\tau_{n-1}(Y_s \circ Y_{s_1}^{-1}/M, Q'')$, as $s$ ranges in
  a dense open subset of $S_0$, is the identity, for an appropriate
  $Q''$. Therefore, the set $\tau_n(Y_s \circ Y_{s_1}^{-1}/M, Q'')$ is
  a one-dimensional set, such that its image under the natural
  projection $\Aut(k[\eps]/(\eps^{n+1}) \to \Aut(k[\eps]/(\eps^{n})$
  is identity, and so, almost coincides with the one-dimensional
  kernel of this projection.
\end{proof}



\subsection{Flat families and intersections}
\label{flat-families}

As already explained, identifying $\Mm$-definably when two curves
(coming from two distinct but fixed definable families) are tangent at
a point $Q\in M^2$ is the key to reconstructing the multiplictative
and additive groups of the field. As we will see, this approach can
only work if we can show that tangency of two definable curves
incident to $Q$ is a non-generic phenomenon. In the present sub-section
we develop the tools allowing us to show that this can, indeed, be
achieved.

Given a strongly minimal family $X \to T$ of $\Mm$-definable plane
curves incident to a point $(Q,Q)\in M^2$ we form the composition
family $X \circ X$ and normalise it to obtain a family $Y \to S$.  Our
aim is to construct an $\Mm$-definable function from $S$ to $T$ taking
$s$ to $t$ if $X_t$ has the same slope at $(Q,Q)$ as $Y_s$. This will
allow us to construct a group configuration based on an
$\Mm$-definable function $T\times T\to T$, which by the results of
Subsection~\ref{sec:composition} corresponds to multiplication in
$K$. The main goal of this Subsection is, therefore, given two
definable families of curves (incident to a fixed point $(Q,Q)$), $X$
and $Y$, to detect $\Mm$-definably and uniformly when a curve $X_t$
and a curve $Y_s$ have the same slope at $(Q,Q)$. We obtain a good
approximation of this goal in Proposition~ \ref{tangency-intersection}
under suitable flatness assumptions. This is one of the main technical
results of the paper.

This strategy is implemented in the present sub-section in a slightly
greater generality, for families of curves in a Cartesian product of
$M^2$ and a variety, and relative intersections, which is necessary
for technical reasons.

We start with recalling a few well-known geometric facts.

\begin{fact}[Generic Flatness, Corollaire~IV.6.11 in \citep{sga1}]
  \label{generic-flatness} Let $Y$ be an integral locally Noetherian
  scheme and let $f: X \to Y$ be a morphism of finite type. Then there
  exists a dense open subset $U \subset Y$ such that the restriction
  of $f$ to $f^{-1}(U)$ is flat.
\end{fact}

\begin{fact}[Local flatness criterion, Proposition~I.2.5 in
\citep{milne}]
% ??? Th\'er\`eme~IV.5.6 in \citep{sga1}.
  \label{flat-criterion} Let $B$ be a flat $A$-algebra and consider $b
  \in B$. If the image of $b$ in $B/\m B$ is not a zero divisor for
  any maximal ideal $\m$ of $A$ then $B/(b)$ is a flat $A$-algebra.
\end{fact}

\begin{fact}[Zariski's Main Theorem, Theorem~1.8 in \citep{milne}]
  \label{zariski-main} If $Y$ is a quasi-compact scheme and $f: X \to
  Y$ is a separated quasi-finite morphism, then $f$ factors as a
  composition $\bar{f} \circ \iota$ where $\bar{f}$ is finite and
  $\iota$ is an open immersion.
\end{fact}

\begin{lemma}
  \label{intersections-are-flat} Suppose that $M$ is a regular curve,
  and $N, N'$ are regular varieties. Let
  $X \subset M^2 \times N \times T$,
  $Y \subset M^2 \times N' \times S$ be two families of
  pure-dimensional curves, and suppose that $X$ is flat over $T$.
  Then the family of scheme-theoretic relative intersections of $X$
  and $Y$ is flat over $T \times S \wo D$ where $D$ is the set of
  pairs $(t,s)$ such that the relative intersection $X_t \cap Y_s$
  infinite.
\end{lemma}

\begin{proof}
  Since flatness is local on the source we may assume that all
  varieties involved are affine.

  Suppose first that $X$ and $Y$ are Zariski closed.  Since regular
  local rings are unique factorization domains, $X$ and $Y$ are
  locally cut out by a single equation, and the relative
  scheme-theoretic intersection of $X$ and $Y$ is a closed
  subscheme of $X \times N' \times S$ that is locally the zero locus
  of a regular function $f$ on
  $M^2 \times N \times N' \times T \times S$ restricted to
  $X \times N' \times S$. By Fact~\ref{flat-criterion}, this closed
  subset is flat precisely over the complement of the subvariety of
  $T \times S$ consisting of those points $(t,s)$ where $f$ does not
  vanish on an irreducible component of $X_t \times \{s\}$. In other
  words, it is flat over the subvariety of points $(s,t)$ where the
  relative intersection $X_t \times_{M^2} Y_s$ is finite.

  In general, since $X$ and $Y$ are families of pure-dimensional
  curves, their relative intersection is dense in the relative
  intersection of their Zariski closures. The statement of the lemma
  follows, since flatness is local on the source.
\end{proof}

\begin{lemma}
  \label{number-sc} Let $f: X \to Y$ be a flat quasi-finite
  morphism. Then the function
  $$
  n: Y \to \Z \qquad y \mapsto \#(f^{-1}(y))
  $$
  is lower semi-continuous, i.e. the lower level sets
  $\suchthat{y}{\#(f^{-1}(y)) \leq n}$ are closed.
\end{lemma}

\begin{proof}
  Follows from (i) of Proposition~15.5.1 of EGA IV.3 \citep{ega} and
  the fact that flat morphisms of finite type are universally open
  (see EGA IV \citep{ega}, 2.4.6).
\end{proof}

\begin{lemma}
  \label{length-sc} Let $f: X \to Y$ be a flat quasi-finite
  morphism. Then, denoting the fibre over a point $y \in Y$ as $X_y$,
  the function
  $$
  l: Y \to \Z \qquad y \mapsto \dim_{k(y)} H^0(X_y, \Oo_{X_y})
  $$
  is lower semi-continuous. If $f$ is finite, then $l$ is locally
  constant.
\end{lemma}

\begin{proof}
  Recall that a finite morphism is projective (EGA II 6.1.11
  \citep{ega}).  Thus, if $f$ is finite the lemma follows from the
  fact that $\dim_{k(y)} H^0(X_y, \Oo_{X_y})$ is the constant term of
  the Hilbert polynomial, and the Hilbert polynomial of a flat
  projective family is locally constant (cf. EGA III 7.9.11).

  In the general case, by Zariski's Main theorem $f$ factors as a
  compostion $\bar f \circ \iota$ where $\bar f: \bar X \to Y$ is
  finite (and hence projective) and $\iota: X \hookrightarrow \bar X$
  is an open immersion. Let $Z_i$ be the connected components of $\bar X
  \wo X$.  By the previous paragraph the function $y \mapsto
  H^0((Z_i)_y, \Oo_{(Z_i)_y})$ is constant on $f(Z_i)$. Therefore
  those lower level sets $\suchthat{y \in Y}{l(y) \leq n}$ that are
  properly contained in $Y$ consist of unions of $Z_i$.
\end{proof}

\begin{lemma}
  \label{length-and-number} Let $Y$ be an irreducible variety and let
  $f: X \to Y$ be a flat quasi-finite morphism. Let $N_1$, $N_2$ be
  the values of the semi-continuous functions $l,n$ of
  Lemmas~\ref{number-sc},\ref{length-sc} on some dense open subset of
  $Y$. Then
  $$
  \suchthat{y \in Y}{ \dim_{k(y)} H^0(X_y, \Oo_{X_y}) < N_1} \subset
  \suchthat{y \in Y}{ \#f^{-1}(y) < N_2}
  $$  
\end{lemma}

\begin{proof}
  Factor $f$ according to Zariski's Main Theorem as the composition of
  a finite $\bar f: \bar X \to Y$ and an open immersion $\iota: X
  \hookrightarrow \bar X$. The number $\dim_{k(y)} H^0(X_y,
  \Oo_{X_y})$ is constant on $\bar X$ by
  Lemma~\ref{length-sc}. Suppose $y$ is such that $\dim_{k(y)}
  H^0(X_y, \Oo_{X_y}) < N_1$, then $f^{-1}(y) \subsetneq \bar
  f^{-1}(y)$ and hence $\#f^{-1}(y) < N_2$.
\end{proof}

Now we can prove the main result of the present sub-section describing
the behaviour of intersection multiplicities in families of
curves. The setting is as follows. We are given a regular curve $M$,
and a point $P=(P_1, P_2) \in M^2$, with a fixed local coordinate
system associated to it, two families of curves
$X \subset M^2 \times N \times T$ and
$Y \subset M^2 \times N' \times S$. In the applications $X$ and $Y$
will be $\Mm$-definable and $\Mm$-stationary, and the first family
will be a obtained as a result of composition of two families of
curves (as in Proposition~\ref{rel-composition}), or by forming an affine
combination of three families of curves (see
Section~\ref{sum-of-correspondences}). We assume, moreover, that all
curves $X_t$ are incident to a fixed point $Q \in M^2 \times N$, and
that all curves $Y_s$ are incident to $Q'$, so that projections of $Q$
and $Q'$ to $M^2$ are both $P$; in the
applications this will be achieved by fixing higher dimensional
families and working with subfamilies passing through the point
$Q$. We want to identify those parameters $t,s$ such that
$\tau_n(X_t/N, Q)=\tau_n(Y_s/N', Q')$. The reason to consider the
relative slope and families of curves in $M^2 \times N$, $M^2 \times N'$ is discussed
in Section~\ref{sec:composition} after the
Definition~\ref{rel-composition-def}.

In the next proposition we show that, in the setting described above,
for each pair $(t,s)$ assumed generic in the parameter variety
$T\times S$, if a curve from $X$ is ``tangent'' to a curve from $Y$ in
the sense that their relative slopes at $Q$ and $Q'$ coincide, then
the number of ``relative intersections'', i.e. points in
$X_t \times_{M^2} Y_s$, drops. This will allow us to approximate
tangency $\Mm$-definably, up to finitely many false positives, which
will be enough for our purposes.

\begin{prop}
  \label{tangency-intersection} 
  Adopt the setting and notation of the previous paragraphs, and pick
  some irreducible top-dimensional components $T_0 \subset T$ and
  $S_0 \subset S$.  Let $n$ be the minimal number such that for
  $(t,s)$ generic in $T_0 \times S_0$ the relative slopes
  $\tau_n(X_t/N, Q)$ and $\tau_n(Y_s/N', Q')$ differ, and let $a$ be
  the natural number such that $\#(X_t\times_{M^2} Y_s) = a$ for all
  $t,s$ in a dense open subset of $T_0 \times S_0$.

  Then there exist dense open $T' \subset T_0$ and
  $S' \subset S_0$ such that  
  \begin{flushleft}
  $\suchthat{ (t,s) \in T' \times S'}{\tau_n(X_t/N, Q) = \tau_n(Y_s/N', Q')
  } \subseteq $
  \end{flushleft}
  \begin{flushright}
    $ \subseteq  \suchthat{(t,s) \in T' \times S'}{\#(X_t \times_{M^2} Y_s) < a}$
  \end{flushright}

\end{prop}

\begin{proof}
  By Fact~\ref{generic-flatness} there exist $T' \subset T_0$,
  $S'\subseteq S_0$ dense open such that $X$ is flat over $T'$ and $Y$
  is flat over $S'$. Let $(X \times S) \times_{M^2} (Y \times T)$ be
  the family, parametrized by $T' \times S'$, of scheme-theoretic
  relative intersections of $X_t$ and $Y_s$ (with possibly non-reduced
  structure), and let $U \subset T' \times S'$ be the set of $(t,s)$
  such that $X_t \times_{M^2} Y_s$ is finite. Let $Z$ denote the 
  preimage of $U$ in $(X \times S) \times_{M^2} (Y \times T)$.

  The variety $Z$ is flat over $U$ by
  Lemma~\ref{intersections-are-flat} and the definition of $U$. Let
  $Q_0$ be the image of $Q \times Q'$ in $M^2 \times N \times N'$. Let
  $Z_0$ be the irreducible component of $Z$ supported at
  $\{Q_0\} \times U$, and let $W$ be the complement of $Z_0$ in $Z$.

  Below we will use subscripts to denote \emph{scheme-theoretic}
  fibres: $Z_{t,s} = Z \otimes k(t,s)$ where $k(t,s)$ is the residue
  field of $(t,s) \in U$. In this notation, we have to show that
  $$
  \suchthat{ (t,s) \in U}{ \tau_n(X_t/N, Q) = \tau_n(Y_s/N', Q')}
  \subseteq \suchthat{ (t,s) \in U}{ \#Z_{t,s} < a} \
  $$
  where $a = \# Z_{t,s}$ for $(t,s) \in U$ generic.

  Let us first prove this statement when the morphism $Z \to U$ is
  finite. Then the number $\dim H^0(Z_{t,s}, \Oo_{Z_{t,s}})$ is
  constant for all $(t,s) \in U$ by Lemma~\ref{length-sc}. Note that 
  $$
  \begin{array}{c}
    \dim H^0( (Z_0)_{(t,s)}, \Oo_{Z_{t,s}}) + \dim H^0( W_{t,s}, \Oo_{Z_{t,s}})
    =
    \dim H^0( Z_{t,s}, \Oo_{Z_{t,s}}) := b \\
     \textrm{ and } \\
    \#Z_{t,s} = \#W_{t,s} - 1\\
  \end{array}
  $$

  For generic $t,s$, $\dim H^0( (Z_0)_{t,s}, \Oo_{(Z_0)_{t,s}}) = n$ and
  therefore $\dim H^0( W_{t,s}, \Oo_{W_{t,s}}) = b - n$.  If the pair
  $t,s$ is such that $\tau_n(X_t/N, Q) = \tau_n(Y_s, Q')$ then
  $$
  \dim H^0((Z_0)_{t,s}, \Oo_{Z_{t,s}}) > n
  $$
  and therefore
  $\dim H^0( W_{t,s}, \Oo_{W_{t,s}}) < b - n$.

  Applying Lemma~\ref{length-and-number} to $W$ we get
  $$
  \suchthat{ (t,s) \in U}{ \dim H^0(W_{t,s}, \Oo_{W_{t,s}}) < b - n} \subseteq
   \suchthat{ (t, s) \in U }{ \#W_{t,s} < a - 1}.
  $$
  The latter set is the same as $\suchthat{ (t, s) \in U }{ \#Z_{t,s} < a}$
  which yields the statement of the Proposition.

  If $Z \to U$ is not finite then compactify it using Zariski's Main
  Theorem: find a finite morphism $\bar Z \to U$ such that $Z$ is an
  open subscheme of $Z$, let $\wbar W$ be the complement of $Z_0$ in
  $\bar Z$; note that $\wbar{W} \wo W = \wbar{Z} \wo Z$. Assume that
  $c = \#\bar Z_{t,s}$,  $d = \dim H^0(\wbar W_{t,s}, \Oo_{\wbar W_{t,s}})$ for
  $t,s$ generic, then we have just shown that
  $$
  \suchthat{ (t,s) \in U}{ \dim H^0(\wbar W_{t,s}, \Oo_{\wbar W_{t,s}} < d -
  n} \subseteq
   \suchthat{ (t, s) \in U }{ \#\bar Z_{t,s} < c}.
  $$
  Observe that 
  $$
  \begin{array}{l}
    \{ (t,s) \in U \ \mid\ \dim  H^0(W_{t,s}, 
    \Oo_{W_{t,s}}) < b - n\} =  \\
	\{(t,s) \in U \ \mid\ \dim H^0(\wbar W_{t,s}, \Oo_{\wbar W_{t,s}}) < d - n\} \cup p(\wbar Z \wo Z) \\
  \end{array}
  $$
   and that
  $$
  \suchthat{ (t,s) \in U}{ \#\wbar Z_{t,s} < c} = \suchthat{ (t,s) \in
    U}{ \#Z_{t,s} < a} \cup p(\wbar Z \wo Z)
  $$
  where $p$ is the the projection of $Z$ onto $U$, which implies
  $$
  \suchthat{ (t,s) \in U}{ \dim H^0(W_{t,s}, \Oo_{W_{t,s}} < b - n} \subseteq
   \suchthat{ (t, s) \in U }{ \#Z_{t,s} < a}.
  $$
  The last inclusion implies the statement of the Proposition.
\end{proof}


\subsection{Interpretation of a one-dimensional group}
\label{first-group}
\begin{thm}
  \label{pure-dim-first-group} Let $\Mm$ be a non-locally modular
  reduct of an algebraic curve $M$ over an algebraically closed field,
  with $X \to T$ a 2-dimensional definable faithful family of
  pure-dimensional curves. Then $\Mm$ interprets a one-dimensional
  group.
\end{thm}

\begin{proof}
  We prove the theorem by constructing a  group
  configuration. We work with almost faithful families of curves
  $Y \to S$ in $M^2 \times N$, where $N$ is some definable set (curves
  in $M^2$ is a particular cas, when $N$ is a point), where $S$ is
  strongly minimal (as a set definable in $\Mm$), subject to
  the requirements that:
  \begin{itemize}
  \item  all curves $Y_s$ are incident to a point $P$
  \item there exists an $\M$-irreducible component $S' \subset S$ of
    dimension 1 such that for all $s \in S'$ the relative slope
    $\tau_n(Y_s/N, P)$ is defined at $P$ for some $n$;
  \item the set of relative slopes $\tau_n(Y_s/N, P)$ of all curves
    with parameter $s$ in $S'$, which is a constructible subset in
    $\Aut(k[\eps]/(\eps^{n+1})$, contains a one-dimensional set 
    almost coinciding with a closed one-dimensional subgroup $H$ of
    $\Aut(k[\eps/(\eps^{n+1})]$.
  \end{itemize}
  
  We construct three such families, incident to points
  $Q_{12}, Q_{21}, Q_{11}$, and relatively incident to $(P_1, P_2)$,
  $(P_2, P_1)$ and $(P_1, P_1)$, respectively, for some
  $P_1, P_2 \in M$, and such that the group $H$ defined above is the
  same for all three families. This will assure that the relative
  slope of the composition of a generic curve in first family with a
  generic curve from the second family can be compared to the relative
  slope of a curve in the third family.

  Let $X \subset M^2 \times T$ be the ample family of plane curves
  given in the assumption. If the base field is of characteristic 0,
  then by Lemma~\ref{many-slopes} there exists a dense open subset
  $U \subset M^2$ such that for any point $Q \in U$, the first-order
  slopes of curves in the family $X^Q \to T^Q$ almost coincide with
  $\Aut(k[\eps]/(\eps^2))$, for a parameter $t$ ranging in some
  irreducible component of $T^Q$.  Pick $P_1,P_2 \in M$ such that
  $(P_1,P_2),(P_2,P_1),(P_1,P_1) \in U$, and consider the families of
  curves $X^{(P_1, P_2)}, X^{(P_2, P_1)}, X^{(P_1, P_1)}$.

  More generally, regardless of the characteristic of the field of
  definition of $M$, by Lemma~\ref{many-slopes2} there exists a family
  of curves $X \subset M^2 \times N \times T$ (with $N$ possibly a
  point), relatively incident to $P_{12}=(P_1, P_2) \in M$, an
  irreducible component $T'$ of $T^P$ such that
  $\{\tau_n(X_t/N, P_{12}): t \in T'\}$ contains a one-dimensional set
  that almost coincides with a one-dimensional subgroup $H$ of
  $\Aut(k[\eps]/(\eps^{n+1})$. By Proposition~\ref{rel-composition},
  the set slopes of the compositions $X_t \circ X^{-1}_s$ almost
  coincides with $H$. Therefore for the purposes of the group
  configuration construction one can take families $X$, $X^{-1}$ and
  $X \circ X^{-1}_t$ for some $t \in T'$ generic enough.

  Let $X \subset M^2 \times N_{12} \times T_{12}$,
  $Y \subset M^2 \times N_{21} \times T_{21}$,
  $Z \subset M^2 \times N_{21} \times T_{21}$ be the three families
  satisfying the requirements stated above, and let
  $W_{12},W_{21},W_{11}$ be the irreducible components of
  $T_{12}, T_{21}, T_{11}$, respectively, making these assumptions
  hold.

  Let $s,t$ be $\M$-independent generic points of $W_{12}, W_{21}$. Note
  that they are generic and independent in the sense of the reduct
  $\Mm$ too. 

  Let $u$ be a point of $W_{11}$ such that
  $$
  \tau_n(Z_u/N_{11}, Q_{11})=\tau_n(Y_s/N_{12}, Q_{12})
  \tau_n(X_t/N_{21}, Q_{21}).
  $$
  Such $u$ exists since relative slopes of $X_t$ and $Y_s$ are generic
  in a one-dimensional subgroup
  $H \subset \Aut(k[\eps]/(\eps^{n+1}))$, so the right-hand side value
  is generic in the same subgroup. Since slopes of curves parametrized
  $W_{11}$ almost coincide with $H$, there exists a generic parameter
  $u \in W_{11}$ fitting the above equality.

  The key to the construction of the group configuration is:\\
   
  \noindent\textbf{Claim:} $u$ is algebraic over $t,s$. 
  \proof Let $P'$ be the the
  image of $Q_{12} \times Q_{21}$ under the composition of maps
  $$
  M^2 \times N \times M^2 \times N' \to M^2 \times N \times_{p_2,
    M^2, p_1} M^2 \times N' \hookrightarrow M^1 \times N \times N'
  $$
  By Proposition~\ref{rel-composition}
  $$
  \tau_n(X_t \circ Y_s/M \times N_{12} \times N_{21}, P') = \tau_n(X_t/N_{12}, Q_{12})\tau_n(Y_s/N_{}, Q_{21})
  $$
  (the relative slope is with respect to the second factor $M$ in
  $M^3$). Let 
  \[l = \#((X_t \circ Y_s) \times_{M^2} Z_w)\] for $w$ generic
  independent from $t$. By Proposition~\ref{tangency-intersection} the $\Mm$-definable set
  $$
  \suchthat{ w \in W }{ \#(X_t \circ Y_s \times_{M^2}
   Z_w ) < l }
  $$
  contains the set of parameters $w \in W$
  such that $\tau_n(Z_w,P) = \tau_n(X_t, P)\tau_n(Y_s, P)$. By the
  very definition this set is finite, and contains $u$, which is what
  we had to show. \qed$_\text{Claim}$\\ 

  In a similar vein, let $a$ be a point of $W_{23}$ independent from
  $t,s$, and let $b \in W_{11}$, $c \in W_{12}$ be points of $W$ such
  that
  $$
  \begin{array}{rcl}
  \tau_n(Z_b/N_{11},
  Q_{11}) & = & \tau_n(X_t/N_{12},Q_{12})\tau_n(Y_a/N_{21},Q_{21}), \textrm{
    and } \\
    \tau_n(X_c/N_{12},Q_{12}) &  = & \tau_n(Y_s/N_{21},Q_{21})\tau_n(Z_b/N_{11},Q_{11})\\
  \end{array}
  $$
  Then
  $$
  \begin{array}{cl}
  & \tau_n(X_c/N_{12},Q_{12})=\tau_n(Y_s/N_{21},Q_{21})\tau_n(Z_b/N_{11},Q_{11})=\\
  = & \tau(Y_s/N_{21},Q_{21})\tau_n(X_t/N_{12},Q_{12})\tau_n(Y_a/N_{21},Q_{21})
  = \\
  = & \tau_n(Z_u/N_{11},Q_{11})\tau_n(Y_a/N_{21},Q_{21})
  \end{array}
  $$
  By the same argument as above, $b$ is algebraic over $a$ and $t$,
  $c$ is algebraic over $s$ and $b$, and $c$ is also algebraic over
  $a$ and $u$, all in the sense of $\Mm$. This implies that
  $$
  \groupconf{t}{s}{u}{a}{b}{\ \ \ \ c}
  $$
  is a group configuration.

  Applying Fact~\ref{hrugrou} we obtain a one-dimensional group
  interpreted in $\Mm$.
\end{proof}



\subsection{Families of curves in reducts of one-dimensional groups}
\label{curves-group}

In the previous sub-section we showed that a non-locally modular reduct
of the full Zariski structure on an algebraic curve $M$ over an
algebraically closed field interprets a strongly minimal group, call
it $H$. The group $H$ is interpretable in an algebraically closed
field and therefore is definably isomorphic to a 1-dimensional
algebraic group by a model-theoretic version of a theorem of Weil on
birational group laws (see \citep{bouscaren-pillay},
\citep{weil-chunk}, \citep{dries-chunk}). Since $\Mm$ is non-locally
modular, so is $H$ (with the full structure induced from $\Mm$). Thus,
by replacing $M$ with $H$, we may assume that $\Mm$ expands a
group. In this setting our goal is to construct a second group
configuration, which will allow us to reconstruct the field in $\Mm$.

Let $H$ be a connected one-dimensional algerbaic group, and let
$Z \subset H^2$ be an irreducible one-dimensional subset that is not a
coset (by non-local modularity there exists an $\Mm$-definable set of
which $Z$ is a component, see \citep{hr-pillay}). We prove that shifts
of $Z$ incident to the identity of $H^2$ have infinitely many distinct
first order slopes at the identity. This will be necessary to
construct a two-dimensional group configuration. In characteristic 0,
an alternative way to obtain a family of curves with enough distinct
first-order slopes is to consider a two-dimensional family of curves
in $H^2$ and apply Lemma~\ref{many-slopes}, then shift the family
obtained to the identity.  Note that the proof of the next lemma is
characteristic-free. In the general case we proceed by a case-by-case
analysis.

\begin{lemma}
\label{aut-cofinite-abelian} Let $A$ be an elliptic curve and $Z$
a closed one-dimensional irreducible subset of $G=A^2$. Identify $T_gG$ with $T_0G$  via the
isomorphism $d\lambda_g: T_0G \to T_gG$,  for $\lambda_g(x)=g \cdot
x$.  Suppose that for any $z \in Z$ the tangent space $T_z Z \subset
T_0G$ is constant. Then $Z$ is a coset of a closed subgroup of $G$.
\end{lemma}

\begin{proof}
  Since $Z$ is a projective curve with a trivial tangent bundle, it is
  an elliptic curve itself. Since any algebraic variety morphism
  between Abelian varieties with finite fibres and preserving the
  identity automatically preserves the group structure by the Rigidity
  Theorem, $Z$ is a coset of an Abelian subvariety of $G$.
\end{proof}

Let $H=\G_a$ or $H=\G_m$. Let $Z$ be a locally closed one-dimensional
irreducible subset of $G=H^2$ such that the restriction to $Z$ of the
projection on the first factor $H$ is \'etale (possibly, after
removing finitely many points). For any $x \in Z$ denote the translate
$$
Y_x := Z\cdot x^{-1}=\suchthat{ u \cdot x^{-1} \in G}{u \in Z}
$$
This defines a family $Y \to Z$ of curves incident to $(e,e)$,
parametrized by $Z$.


\begin{lemma}
  \label{aut-cofinite-ga-gm} In the setting as above, assume $H$ is
  defined over a field of characteristic 0. Fix a local coordinate
  system at the identity point $e \in H$ so that the slope of any
  curve incident to $(e,e)$ is well-defined.  Suppose that the slope
  $\tau_1(Y_x,(e,e))$ is constant for $x$ in an open neighbourhood of
  $(e,e)$. Then $Z$ coincides with an open subset of a closed subgroup of
  $G$.
\end{lemma}

\begin{proof}
  Without loss of generality we may assume that $(e,e) \in Z$.

  Let $H=\G_m$, and let $Z$ be cut out by an equation $h(x,y)=0$. Then
  $Y_{(x_0, y_0)}$ is cut out by the equation $h(x \cdot x_0, y \cdot
  y_0)$, and 
  $$
  \tau_1(Y_{(x_0,y_0)}, (e,e))=\left.\dfrac{\partial_x h(x\cdot x_0,y\cdot
        y_0)}{\partial_y (x\cdot x_0,y\cdot y_0)}\right|_{x=1, y=1} =
  \dfrac{\partial_x h (x_0,y_0) \cdot x_0}{\partial_y
      h (x_0,y_0) \cdot y_0}
  $$  
  Therefore, if $f$ is the expansion of $Z$ into formal power series
  at the identity then by Proposition~\ref{slope}
  $\dfrac{\partial_x h(x,f)}{\partial_y h(x,f)}$ is $f'$, the formal
  derivative of $f$. Since we are working in a local coordinate system
  picked at $(e,e)$, $x_0=x+1$ and $y_0=f+1$, so if
  $\tau_1(Y_x, (e,e))$ is constant for $x$ in the neighbourhood of
  $(e,e)$, then for some $a\in k$ the formal power series $f$
  satisfies the differential equation
  $$
  f'\cdot \dfrac{x+1}{f+1} = a
  $$
  
  Similarly, for the additive group ($H=\G_a$) 
  $$
  \tau_1(Y_{(x_0,y_0)}, (e,e)) = \left.\dfrac{\partial_x h(x + x_0,y +
        y_0)}{\partial_y h(x + x_0,y + y_0)}\right|_{x=0, y=0} =
  \dfrac{\partial h_x(x_0,y_0)}{\partial_y h (x_0,y_0)}
  $$
  and the corresponding differential equation is
  $$
  f' = a
  $$
  
  The series $f = a x$ satisfies the second equation, and by
  Lemma~\ref{ode-unique} this is the only solution with zero constant
  term. It follows that $Z$ is defined by the equation $y = ax$, and
  so is a subgroup of $\G_a \times \G_a$.

  In case $H=\G_m$ it follows from
  Lemma~\ref{solutions-slope-gm-char0} that  $f=(x+1)^a - 1$ is the
  unique formal power series solution of the differential equation
  $$
  y' = a  \dfrac{f+1}{x+1}
  $$
  and is only integral over $k[x]$ if $a$ is rational, in which case
  $f$ is a formal power series expansion at $(1,1)$ of an irreducible
  component of a curve defined by an equation $y^n=x^m$. But all such
  curves are subgroups of $\G_m \times \G_m$.
\end{proof}

\begin{lemma}
  \label{many-slopes-ga-charp} In the same setting, suppose that
  $H=\G_a$ is defined over a field of positive characteristic.
  Suppose that $Z$ is not a coset of a subgroup of $G$ and that the
  projection of $Z$ on the first factor $H$ is \'etale on some dense
  open set. Then there exists a definable family of curves relatively
  incident to $(0,0) \in \G_a \times \G_a$ such that the set of
  relative first order slopes at $(0,0)$ almost coincides with
  $\Aut(\Spec k[\eps]/(\eps^2))$.
\end{lemma}

\begin{proof}

  Let $W \subset \G_a \times \G_a \times N$ be a curve, where $N$ is a
  Cartesian product of finitely many copies of $\G_a$, and assume that
  $(0,0,\ldots,0) \in W$. For any $t \in p(W)$, where $p$ is the
  projection on $\G_a^2$, let $W_t$ denote the shift by $t$:
  $$
  W_t := \suchthat{ (x - t, y - t, z)}{ (x,y,z) \in W}
  $$
  Let $f_t$ be the relative formal power series expansion of $W_t$ at
  $(0,0, \ldots, 0)$. Let $W_0$ be the irreducible component of $W$
  containing $0$. 

  The proof is based on the iterated application of the following
  observation, that essentially uses the fact that $W$ is not a
  coset:\\ 

  \noindent\textbf{Claim}.  There are two mutually exclusive
  possibilities:
  \begin{itemize}
  \item[-] $f_t = ax + (g_t)^{p^n}$ for some $g_t \in xk[[x]]$
    depending on $t$, and so $\tau_1(W_t/N, (0,0,\ldots, 0)$ is
    constant for $t$ generic in $W_0$.
  \item[-] the set of relative first order slopes
    $\tau_1(W_t/N, (0,0,\ldots, 0)$ as $t$ varies in $W_0$ almost
    coincides with $\Aut(k[\eps]/(\eps^2))$.
  \end{itemize}

  \proof Suppose the second statement is not the case, and
  $\tau_1(W_t/N, (0,0,\ldots, 0)$ takes a single value $a$.  By the
  same argument as in the proof of Lemma~\ref{aut-cofinite-ga-gm} $f$
  satisfies the differential equation
  $$
  f' = a
  $$
  The solutions of the differential equation are of the form
  $a x + h^{p^n}$, where $h \in (x) \subset k[[x]]$, by direct
  observation. If $f = ax$ then $p(Z)$ is a coset, contradicting our
  assumption. \qed$_\text{Claim}$\\
  
  Let $l$ be the largest integer such that $\tau_l(W_t/N, (0,0))$
  depends non-trivially on $t$. If $l = 1$, we are done. Otherwise,
  since $W$ is not a coset, by the Claim $f_t = ax + (g_t)^{p^n}$,
  where $g_t \in (x) \subset k[[x]]$ and $n$ is maximal for $f_t$ to
  have such form. Consider $W'_t := W_t - W_{t_0}$ for some $\M$-generic
  $t_0$. By Proposition~\ref{addition} the relative formal power
  series expansion of $W'_t$ at $(0,0,\ldots, 0)$ is $h_t^{p^n}$, where
  $h_t=g_t - g_{t_0}$.

  Then $h_t$ is a relative formal power series expansion of
  $W''_t:=\Fr^{-n}_{\G_a^2 \times N/\G_a \times N}(W'_t)$ at $(0,0, \ldots,
  0)$. Pick some $t_1$ such that $h_{t_1} \neq 0$. Then by the claim
  either $h_{t_1} = bx + (h'_{t_1})^{p^n}$, $b \neq 0$, or shifts of
  $W''_{t_1}$ have infinitely many relative first order slopes at
  $(0,0,\ldots,0)$.

  In the latter case note that by Lemma~\ref{cancel-out-pth-power},
  $\tau_1((W''_{t_1} - t) \circ (W''_{t_0})^{-1}/N,
  (0,0,\ldots,0))=\tau_1((W'_{t_1} - t) \circ W'^{-1}_{t_0},
  (0,0,0,\ldots,0)$, and so the family of curves
  $(W'_{t_1} - t) \circ (W'_{t_0})^{-1}$ satisfies the requirements of
  the lemma.

  In the former case, the relative formal power series expansion of
  $W''_{t} \circ W''^{-1}_{t_1}$ is well-defined, and is of the form
  $x + (h''_t)^{p_n}$ by Lemma~\ref{cancel-out-pth-power}. Clearly,
  the smallest number $l'$ such that
  $\tau_{l'}(W'_t \circ (W'_{t_1})^{-1}/N, (0,0,\ldots,0))$ is
  non-constant for generict $t$ in $W_0$, is strictly less than $l$,
  so we may run the proof again on the definable curve
  $W'_t \circ (W'_{t_1})^{-1}$. After finitely many iterations, $l$
  must be 1.
\end{proof}

\begin{lemma}
  \label{many-slopes-gm-charp} In the same setting, suppose that
  $H=\G_m$ is defined over a field of positive characteristic.  Suppose
  that $Z$ is not a coset of a subgroup of $G$ and that the projection
  of $Z$ on the first factor $H$ is \'etale on some dense open
  set. Then there exists a family of curves incident to a point $Q\in
  \Delta_M$ such that the set of first order slopes at $Q$ almost
  coincides with $\Aut(\Spec k[\eps]/(\eps^2))$.
\end{lemma}

\begin{proof}
  The proof of Lemma~\ref{many-slopes-ga-charp} goes through unaltered
  applying the following Claim instead of the claim proved there, and
  with the main step of induction (passing from a family of curves
  with formal power series expansion $ax + g_t^{p^n}$ to a family of
  curve with formal power series expansions $h_t^{p^n}$) replaced with
  a similar argument proper to $\G_m$. As in
  Lemma~\ref{many-slopes-ga-charp}, we consider the family of
  translates $W_t$ of the curve $Z$, and denote the formal power
  series expanson of a shift by a point $t \in Z$ as $f_t$.

  \noindent\textbf{Claim}.  There are two mutually exclusive
  possibilities:
  \begin{itemize}
  \item[-] $f_t = (1 + g_t^{p^n})(1+x)^a - 1$ for some
    $g_t \in (x) \subset k[[x]]$ depending on $t$, and
    $\tau_1(W_t/N, (0,0,\ldots, 0)$ is constant for $t$ generic in
    $W_0$.
  \item[-] the set of relative first order slopes
    $\tau_1(W_t/N, (0,0,\ldots, 0)$ as $t$ varies in $W_0$ almost
    coincides with $\Aut(k[\eps]/(\eps^2))$.
  \end{itemize}
  
  \proof By the same reasoning as in Lemma~\ref{aut-cofinite-ga-gm} the
  formal power series expansion of a curve $Z$ in $\G_m \times \G_m$
  such that all their shifts have a constant first order slopes
  satisfies the equation
  $$ 
  f' = a\cdot \dfrac{1+f}{1+x}
  $$
  for some element $a \in k$.

  By Lemma~\ref{solutions-slope-gm-charp} the solutions of the
  differential equation are of the form $g(1+x)^a -1$ where
  $g \in 1+ (x^{p^n}) \subset k[[x]]$. If $f = (1+x)^a - 1$ for
  $a \in \F_p$ then $Z$ is the curve defined by $y = x^a$ which
  contradicts our assumption that $W$ is not a coset of a
  subgroup.\qed$_\text{Claim}$\\ 

  If we are in the second case of the Claim, we are done. Otherwise,
  as in the proof of Lemma~\ref{many-slopes-ga-charp}, we replace the
  family of translates by the family $W_t-W$ and show, that after
  finitely many such replacements, we will necessarily end with a
  family as in the conclusion of the lemma. The inductive step is
  different than in the proof of Lemma~\ref{many-slopes-ga-charp},
  because of a different formal group law (multiplicative, not
  additive).

  If $W_1$ and $W_2$ are two curves with formal power series
  expansions, $h_1 := (1 + g_1^{p^n})(1+x)^a -1$ and
  $h_2 := (1 + g_2^{p^n})(1+x)^a -1$, with $g_1, g_2 \in (x)$ then by
  Proposition~\ref{addition}, $Z_1 - Z_2$ (here, as elsewhere in the
  paper, we use additive notation when referring to the ``sum of
  curves'' operation from Definition~\ref{relsum-defn}, with respect
  to the \emph{multiplicative} group law) has (relative) power
  series expansion $\dfrac{h_1 + 1}{h_2 + 1} - 1$ by
  Proposition~\ref{expansion-is-point}, and so is
  $$
  \dfrac{((1+g_1^{p^n})(1+x)^a - 1) + 1}{((1+g_2^{p^n})(1+x)^a - 1) +
    1} - 1 = \dfrac{1+g_1^{p^n}}{1+g_2^{p^n}} - 1
  $$
  which clearly belongs to $(x^{p^n}) \subset k[[x^{p^n}]] \subset
  k[[x]]$.
\end{proof}


\subsection{Interpretation of the field}
\label{second-group}

With the preparations made in the previous section we can now show
that a non-locally modular reduct of one-dimensional algebraic group
interprets a field. This strengthens the main result of
\citep{marker-pillay}.

\begin{thm}
  \label{marker-pillay-plus}
%  \label{field-interp}
  Let $\Mm=(G,\cdot,\ldots)$ be a reduct of a 1-dimensional algebraic
  group $(G,\cdot)$ over an algebraically closed field of any
  characteristic, which is not locally modular. Then $\Mm$ interprets a
  field.
\end{thm}


\begin{proof}
  By Lemma~\ref{aut-cofinite-abelian}, Lemma~\ref{aut-cofinite-ga-gm},
  Lemma~\ref{many-slopes-ga-charp} or Lemma~\ref{many-slopes-gm-charp}
  there exists a definable family
  $Y \subset G \times G \times N \times S$, where $N$ is a Cartisian
  product of several copies of $G$, of curves incident to
  $P=(e,e,\ldots, e)$ and an irreducible set $S_0 \subset S$ such that
  for $s$ generic in $S_0$, the relative slope
  $\tau_1(Y_s/N, (e,e,\ldots, e)$ is not constant.

  Take $a_1,a_2,b_1,b_1, u \in S_0$ generic and pairwise independent
  (in the sense of the reduct structure $\Mm$). Let $c_1,c_2$ be such
  that (all slopes are taken at $P$)
  \begin{eqnarray*}
  \tau_1(Y_{c_1}/N, P) & = & \tau_1(Y_{a_1}/N, P) \tau_1(Y_{b_1}/N, P)\\
  \tau_1(Y_{c_2}/N, P) & = & \tau_1(Y_{a_2}/N, P) \tau_1(Y_{b_1}) +
                             \tau(Y_{b_2}/N, P)\\
  \end{eqnarray*}
  Since the image of the function $x \mapsto \tau_1(Y_x, P)$ for $x$
  ranging in $S_0$ is one-dimensional the values of expressions in the
  right-hand-side are generic in $\End(\Spec k[\eps]/(\eps^2))$ for
  generic values of parameters. Therefore
  $$
  \tau_1(Y_{a_1}/N, P) \tau_1(Y_{b_1}/N, P) \textrm{ and }
  \tau_1(Y_{a_2}/N, P) \tau_1(Y_{b_1}/N, P) + \tau_1(Y_{b_2}/N, P)
  $$
  are generic, and $c_1, c_2$ with such slopes can be picked in $S_0$.

  Let $z, v$ be such that
  \begin{eqnarray*}
  \tau_1(Y_z/N, P) & = & \tau_1(Y_{a_1}/N, P) \tau_1(Y_u/N, P) +
                         \tau_1(Y_{a_2}/N, P)\\
  \tau_1(Y_v/N, P) & = & \tau_1(Y_{b_1}/N, P)^{-1} \tau_1(Y_u/N, P) -
                         \tau_1(Y_{b_2}, P) \\
  \end{eqnarray*}
  By a similar reasoning, $z, v$ are generic. It also follows form the
  way $c_1, c_2, z, v$ were defined that
  $$
  \tau_1(Y_z/N, P)  =  \tau_1(Y_{c_1}/N, P) \tau_1(Y_v/N, P) +
  \tau_1(Y_{c_2}/N, P)
  $$

  We will now show that $(c_1, c_2)$ is algebraic over $(a_1, a_2)$
  and $(b_1, b_2)$ in the sense of $\Mm$. Denote $P'$ and $P''$ points
  in $G^3 \times N \times N$, repectively in
  $G^4 \times N \times N \times N$, that project to identity on every
  factor. By Proposition~\ref{rel-composition}, the remark before
  Proposition~\ref{addition}, and Lemma~\ref{relslope-proj}
  \begin{eqnarray*}
    \tau_1(Y_{a_1} \circ Y_{b_1}/G \times N \times N, P') & = & \tau_1(Y_{a_1}/N, P) \tau_1(Y_{b_1}/N, P),\\
    \tau_1(Y_{a_2} \circ Y_{b_1} + Y_{b_2})/G^2 \times N \times N
    \times N, P'') & = & 
    \tau_1(Y_{a_2}/N, P) \tau_1(Y_{b_1}/N, P) + \tau_1(Y_{b_2}/N, P)\\
  \end{eqnarray*}
  
  Let
  $l_1 = \#(Y_{c_1} \times_{G^2} Y_{a_1} \circ Y_{b_1}), l_2 =
  \#(Y_{c_2} \times_{G^2} Y_{a_2} \circ Y_{b_1} + Y_{b_2})$ for
  $a_1, a_2$, $b_1, b_2$, $c_1, c_2 \in S_0$ generic and
  independent. Since the number of intersections is a first-order
  property, it does not matter what particular parameters
  $a_i, b_i, c_i$ we take as long as they are generic and independent
  (in the sense of the reduct structure $\Mm$).
  By Proposition~\ref{tangency-intersection} the $\Mm$-definable set
  $$
  \suchthat{w \in S_0}{ \#(Y_w  \times_{G^2} (Y_{a_1} \circ Y_{b_1})) < l_1}
  $$
  contains $c_1$ and by definition of $l_1$ is finite. By
  Proposition~\ref{tangency-intersection} again the $\Mm$-definable
  set
  $$
  \suchthat{w_2 \in S_0}{ \#(Y_w \times_{G^2} (Y_{a_2} \circ Y_{b_1} + Y_{b_2})) < l_2}
  $$
  contains $c_2$ and by definition of $l_2$ is finite.  

  Arguing in a similar fashion, by application of
  Proposition~\ref{tangency-intersection}, we deduce that 
  \begin{itemize}
  \item $c_1$ and $c_2$ are algebraic over $z,v$,
  \item $a_1$ and $a_2$ are algebraic over $z,u$,
  \item $b_1$ and $b_2$ are algebraic over $v,u$
  \end{itemize}
  in the reduct $\Mm$.

  It follows from the discussion above that
  $$
  \groupconf{\hspace{-2em}(a_1, a_2)}
  {\hspace{-2em}(b_1, b_2)}
  {\hspace{-2em}(c_1, c_2)}{z}{u}
  {\hspace{1em}v}
  $$    
  constitutes a group configuration. Therefore, by Fact~\ref{hrugrou}
  there exists a two-dimensional group definable in $(G,\cdot,Z)$ that
  acts transitively on a one-dimensional set. One checks that the
  conditions of the Fact~\ref{hrugrou-faithfulness} are verified as
  well.  By Fact~\ref{fieldconf}, the group $G$ is isomorphic to
  the affine group $\G_a(k) \rtimes \G_m(k)$ of an infinite definable
  field $k$.
\end{proof}


\begin{thm}
  \label{pure-dim-field}
  Let $\Mm$ be a non-locally modular reduct of an algebraic curve $M$
  over an algebraically closed field $M$, that has a definable
  faithful two-dimensional family $X \to T$ of pure-dimensional
  curves. Then $M$ interprets a field.
\end{thm}

\begin{proof}
  Conjunction of Theorem~\ref{pure-dim-first-group} and
  Theorem~\ref{marker-pillay-plus}. 
\end{proof}




\section{Getting rid of zero-dimensional components}
\label{pure-dimensional}

In the previous section we defined a group and then a field in $\Mm$
given a sufficiently large definable family of plane curves, $X$,
whose generic fibres have no $0$-dimensional connected components. The
aim of this section is to construct such a family in $\Mm$.

The distinction between genericity in the sense of the reduct, $\Mm$,
and in the sense of the full Zariski structure $\mathfrak M$ will be
of importance in the some of the subtler arguments of the present
section. To distinguish between the two we use \emph{reduct generic}
(or $\Mm$-generic) for the former, and \emph{generic},
$\mathfrak M$-generic or \emph{field generic} for the latter. Note
that $\mathfrak M$-genericity implies $\Mm$-genericity but not, a
priori, vice versa.

Drawing upon a tradition in the model-theoretic literature,
one-dimensional definable sets in $M^2$ will be refereed to as ``plane
curves''.


\subsection{Preliminaries}
\label{outline}

For a 1-dimensional definable set, $Z$ , denote $Z^1$ the union of its
1-dimensional connected components, $Z^0$ the union of its
0-dimensional components, $\wbar{Z^1}$ the closure of $Z^1$ and
$\Fr(Z)=\wbar{Z^1} \wo Z^1$ . The same notation will apply for
families: if $Y \to S$ is a family of one-dimensional sets, then e.g.
$\Fr(Y) \to S$ is the family consisting of frontiers of
elements of the family $Y$.

The results of this section use only basic intersection theory and
are, to a large extent, independent from previous sections. Our main
results (stated in somewhat greater generality than we actually need)
is: 

\begin{thm}\label{cleanup-general}
  Let $M$ be an algebraic curve over an algebraically closed field
  $k$. Let $\Mm$ be a strongly minimal non-locally modular reduct of
  the $k$-induced structure on $M$. Let $S\subseteq M^2$ be an
  $\Mm$-definable strongly minimal set. Then $S^0\subseteq
  \acl_\Mm([S])$, where $[S]$ is a canonical parameter for $S$.  
\end{thm}

This theorem follows from the following, somewhat more technical
result:  

\begin{prop}\label{cleanup}
  Let $\Mm$ be as above. Let $X\to T$ be a faithful $\Mm$-definable
  family of curves with $\dim(T)\ge 3$. Assume, moreover, that
  if $t\in T$ is generic and $P\in X_t^0$ then $P$ is generic over $\0$. Then
  there exists an $\Mm$-definable faithful family of plane curves
  $\widetilde{X}\to T$ such that for all generic $t\in T$:  
  \begin{enumerate}
  \item $X_t\sim \widetilde{X}_t$. 
  \item $\widetilde{X}_t$ is pure-dimensional.  
  \end{enumerate}
\end{prop}

Our strategy is as follows. First, we show the existence of an
$\Mm$-definable family of plane curves $X\to T$ satisfying all the
technical assumptions of the previous proposition. Fixing $s\in T$
field generic and $P\in X_s^0$ our assumptions assure that any generic
independent $t,u\in T^P$ are, in fact, generic independent over
$\0$. Assuming, as we may, that $T$ is $\Mm$-stationary it follows
that $ \#(X_t \cap X_u)$ is independent of the choice of
$t,u$. Moreover, we show that
$\#(X_t \cap X_u)= \#(\overline {X^1_t} \cap
\overline{X^1_u})$. Assuming towards a contradiction that
$P\notin \acl_\Mm(s)$ we get immediately that
$\#(X_t \cap X_u)=\#(X_t \cap X_s)$, implying -- as $P$ is isolated in
$X_s$ -- that
$\#(\overline {X^1_t} \cap \overline{X^1_u})> \#(\overline {X^1_t}
\cap \overline{X^1_s})$. We then apply basic intersection theory to
show that, as $t$ was arbitrary, this leads to a contradiction.

We start by addressing the technical requirements Proposition
\ref{cleanup}. This will require a few steps. The first result we need
is well known to the experts, and goes back to Hrushovski's PhD thesis
and Buechler's works from the early 1980s (see e.g.,
\cite[p.88]{hrushovski1989unidimensional}). As we were unable to find
an explicit reference, we give a brief overview of the proof:

\begin{lemma}
  \label{big-generic-family}
  If $\Mm$ is a non-locally modular strongly minimal set there is no
  bound on the dimension of definable families of plane curves.  
\end{lemma}

\begin{proof}
  Clearly, if a field is interpretable in $\Mm$ then the lemma is
  true. It will suffice, therefore, to prove that if $Z\to S$ is an
  $n$-dimensional ($n\ge 2$) faithful family then either
  $\dim(Z\circ Z)>n$ or a field is interpretable in $\Mm$.
    
  Let $S'$ be a parameter set for the normalisation of $Z\circ Z$. We
  prove that if $\dim S' = n$ then $\Mm$ interprets an infinite
  field. Our assumption implies that there exists an $\Mm$-definable
  finite-to-finite correspondence $\mu: S' \vdash S$. By definition,
  there is also an $\Mm$-definable function $p:S\times S\to S'$
  (defined by the requirement that $Z_t\circ Z_s$ is -- up to a finite
  set -- the curve defined by $p(t,s)$).
  
  Let $t,s$ be $\Mm$-generic independent elements of $S$ and 
  $u\in \mu(p(t,s))$. Let $x$ be a reduct generic point of $M$, and
  $y \in Z_t(x)$, $z \in Z_s^{-1}(x)$. Construct the following
  configuration:
  $$
  \groupconf{t}{s}{u}{y}{x}{z}
  $$
  As $y \in Z_t \circ Z_s(z)$ it follows that  $y$ is
  inter-$\Mm$-algebraic with $x$ over $u$, implying that the above is, in
  fact, a group configuration.  By Fact~\ref{hrugrou} there exists an
  $\Mm$-definable group $G$ of dimension $n$ acting definably on a definable
  set $X$ of dimension 1.  The canonical base of $\tp(y,x/t)$ is
  inter-algebraic with $t$ by faithfulness of the family $Z$, and
  similarly for the canonical bases of $\tp(z,x/s)$ and
  $\tp(z,x,/u)$. Therefore by Fact~\ref{hrugrou-faithfulness}
  the action of $G$ on $X$ is faithful. By Fact~\ref{fieldconf}
  there exists a field definable in $\Mm$.  
\end{proof}

In the discussion that follows canonical parameters (see Sub-section
\ref{sec:parameters}) will play an important role. We denote, given a
definable set $S$, its canonical parameter $[S]$. Formally, we have to
distinguish between $\Mm$-canonical parameters and field-canonical
parameters. In practice, and in order to overload the notation, we
will always use $\Mm$-canonical parameters (as long as the definable
sets in questions are $\Mm$-definable, of course).

Now we turn to the genericity of isolated points (in the sense of the
assumptions of Proposition \ref{cleanup}):  

\begin{lemma}
  Let $S\subseteq M^2$ be an $\Mm$-definable curve with
  $[S]\notin \acl_\mathfrak{M}(\0)$. Let $Z\to T$ be an $\Mm$-irreducible
  $n$-dimensional $\Mm$-definable faithful family of plane curves for
  some $n\ge 2$. Then there exists an $\Mm$-definable family of plane
  curves $Z'\to T$ with $Z'_t\sim Z_t$ for all $t\in T$ and such that
  for any $t\in T$ generic over $[S]$ any point in $(S\circ Z'_t)^0$
  is field generic.
\end{lemma}

\begin{proof}
  We may assume that $S$ is $\Mm$-stationary (otherwise, repeat the
  argument for each $\Mm$-component of $S$ separately) and projecting
  dominantly on both coordinates. We may also assume that there is no
  point $P\in M^2$ incident to $Z_t$ for all generic $t$.  We may
  assume that if $(p,q)\in S^0$ then $p,q\notin \acl_{\mathfrak M}(\0)$, as if that
  is not the case we can replace $Z$ with
  $Z\setminus (\{p\}\times M\times T\cup M\times\{q\}\times T)$.
	
  Now note that if $(p,q)\in S^0$ then $p,q\in \acl_\mathfrak{M}([S])$. Thus, if
  $t\in T$ is $\M$-generic over $[S]$ then any point of the form $(p,r)$,
  where $(q,r)\in Z_t$, is $\M$-generic over $\0$. Indeed, since $q$ is
  generic over $\0$ and $t$ is generic over $q$ any point
  $(q,r)\in Z_t$ is $\M$-generic in $Z_t$, therefore $\M$-generic over $\0$.
	
  If for some $(p,q)\in Z_t^0$, say, $p\in \acl_\M(\0)$ then, for all
  $\M$-generic $t'$ in the same irreducible component of $T$ there
  exists $q'$ such that $(p,q')\in X_{t'}^0$. Since $T$ has finitely
  many components there are $\{p_1,\dots, p_k\}\subseteq \acl_\M(\0)$
  such that whenever $(p,q)\in Z_t^0$ with $p\in \acl_\M(\0)$ for an
  $\M$-generic $t$ there is some $i$ such that $p=p_i$. Similarly,
  there are $\{q_1,\dots, q_r\}$ such that whenever $t\in T$ is
  $\M$-generic and $(p.q)\in X_t^0$ with $q\in \acl_\M(\0)$ there is
  some $i$ such that $q=q_i$.
  
  
  Setting $Z'\to T$ by 
  \[
    Z'=Z_t\setminus \bigcup\limits_{i=1}^k \{p_i\}\times 
    M \times T\cup \bigcup\limits_{i=1}^r M\times \{q_i\}\times T 
  \]
  we may assume that for $\M$-generic $t\in T$ and any $(p,q)\in Z_t^0$
  both $p$ and $q$ are $\M$-generic (not necessarily independent) over
  $\0$.  
	
  Because $[S]\notin \acl_\M(\0)$ for generic $(a,b)\in S$ we have
  $\dim(a,b)=2$. It follows from our assumptions on $Z$ that if $t\in
  T$ is $\M$-generic over $[S]$ then for $(p,q)\in Z_t^0$ and $r$ such
  that $(r,p)\in S$, we have $\dim(r,p)=2$. If $\dim(q/p)=0$ this
  implies  
  \[\dim(p,q,r)=\dim(r,q)+\dim(p/r,q)=\dim(r,q)=2\] 
  and if $\dim(q/p)=1$ then $\dim(r/p,q)=1$ because $t$ is
  $\M$-generic over $[S]$. In any case we get that $\dim(r,q)=2$. As
  $(S\circ Z_t)^0\subseteq S^0\circ Z_t\cup S\circ   Z_t^0$, the
  conclusion of the lemma follows. 
\end{proof}

We may now conclude: 

\begin{cor}\label{cleanup-assumptions}
  There exists an $\Mm$-definable family of plane curves $X\to T$
  satisfying the assumptions of Proposition \ref{cleanup}. 
\end{cor}

\begin{proof}
  Fix $Z\to T$ a faithful $\Mm$-definable family of plane curves
  of dimension at least $3$, as provided by non local
  modularity. Fix an $\M$-generic $Z_{t_0}$ in that family. By the
  previous lemma, we can find $Z'\to T$ of the same dimension
  such that $X:=Z_{t_0}\circ Z'$ has the desired properties. 
\end{proof}

\noindent\textbf{Notation} We fix a
family $X\to T$ satisfying the assumptions of
Proposition~\ref{cleanup}. As the proof proceeds we may replace $X$
with other families (such as $X\circ X^{-1}$) satisfying stronger
assumptions. In order not to overload the notation, we will be
explicit about any such replacement, and will still denote the
resulting family $X\to T$.

\subsection{Indistinguishable points}\label{sec:ind}



Our aim is to use intersection theory in order to identify the
isolated components of $X_t$ for $t\in T$ generic. Our setting,
however, only allows us direct access to global intersection
properties (such as the number of geometric intersection points of two
curves), and for such global phenomena the existence of isolated
points, frontier points and other local obstructions of similar
flavour, may interfere with the geometric argument. We now turn to
studying the nature of these possible obstructions.

\begin{defn}
  Let $X\to T$ be a definable family of plane curves, $P\in M^2$ any
  point. A point $Q\neq P$ is \emph{$X$-indistinguishable} from $P$ if $T^P\sim T^Q$. The point $Q$ is
  \emph{frontier-$P$-indistinguishable} (with respect to a field
  generic type $p$ extending  $X$) if $Q\in \Fr(X_t)$ for all (field)
  generic $t\in T^P$ such that $t\models p$.  
\end{defn}

Note that for an $\Mm$-definable family $Y$ of plane curves the
property of being $Y$-indistinguishable is
$\Mm$-definable, while the property of being frontier
indistinguishable is, a priori, only definable in the full Zariski
structure on $M$.

\begin{rem}
  In the definition of frontier indistinguishable points (and in all
  further references to frontier point in the present section) we have
  intentionally omitted any clear reference to the topological space
  where this frontier is computed. In the algebro-geometric context of
  the present text this has no importance (in the proof of Proposition 6.2 we will consider an open embedding of $M$ into a regular proper curve, $M_2$, and frontier will always refer to the ambient Zariski topology). In other contexts where one
  may consider generalising the results of this section the main
  requirement to keep in mind is that the frontier of a plane curve be
  finite.
\end{rem}

There are known examples, due to Hrushovski (see, e.g., \cite{Martin})
of strongly minimal reducts of algebraically closed fields where to
any generic point, $P$, there exists $Q\neq P$ interalgebraic with $P$
over $\0$  and indistinguishable (with respect to any $\0$-definable
family of plane curves) from $P$. In the present sub-section we show
how indistinguishable points can be avoided: 
\begin{prop}\label{cleanup-ind}
  Let $X\to T$ be an $\Mm$-definable faithful $\Mm$-stationary family
  of plane curves. Then, after possibly removing finitely many points
  from $M$:
  \begin{enumerate}
  \item If there are no generic $X$-indistinguishable points, then
    there are no generic $X\circ X^{-1}$-indistinguishable points.
  \item There exists an $\Mm$-definable equivalence relation $\nsim$
    on $M$ such that there are no
    $(X/\nsim)\circ (X/\nsim)^{-1}$-indistinguishable points in
    $M/\nsim\times M/\nsim$, where
    \[X/\nsim\to T:=\{([x],[y],t): t\in T, (x,y)\in X_t\}.\]
  \end{enumerate}
\end{prop}
	
\begin{proof}
  Let us denote $XX:=X\circ X^{-1}$. The proof proceeds by a series of claims. The key observation is: \\
	
  \noindent\textbf{Claim I:} Suppose that any generic $P\in M^2$
  has some $Q\in M^2$ that is $XX$-indistinguishable from it. Then for
  all generic $r$ there is $s$ such that $(p_1,r)$ is
  $X$-indistinguishable from $(q_1,s)$ and $(p_2,r)$ is
  $X$-indistinguishable from $(q_2,r)$. In particular
  $q_1\in \acl_\Mm(p_1)$.
	
  \proof Let $P=(p_1,p_2)$ and $Q=(q_1,q_2)$ be generic and
  $XX$-indistinguishable. Fix any $r\in M$, and any generic (over all
  the data) $t\in T^{(p_1,r)}$. Let $S:=X_t(q_1)$. Then $S$ is
  finite. The assumption that $P$ is $XX$-indistinguishable from $Q$
  implies that any generic curve through $(p_2,r)$ also passes through
  $(q_2,s)$ for some $s\in S$. Since $S$ is finite and $T^{(p_2,r)}$
  is stationary there is $s\in S$ such that any generic curve through
  $(p_2,r)$ passes through $(q_2,s)$. In other words, $(p_2,r)$ is
  $X$-indistinguishable from $(q_2,s)$.  Since $r$ was generic, this
  implies that for all generic $r$ there exists $s$ such that
  $(p_2,r)$ is $X$-indistinguishable from $(q_2,s)$.
	
  So fix a generic $u$ such that $(p_2,r)\in X_u$. Then
  $(q_2,s)\in X_u$. By the exact same argument as above, we get that
  for any generic enough $w\in T^{(p_1,r)}$ we get $(q_1,s)\in
  X_w$. I.e., $(p_1,r)$ is $X$-indistinguishable from $(q_1,s)$. This
  implies that $q_1$ is algebraic over $p_1$. For otherwise, any
  generic point $(p_1,r)$ would have infinitely many points
  $X$-indistinguishable from it, contradicting the faithfulness of the
  family. \qed$_\text{Claim I}$\\
	
  From this we can immediately conclude the first part of the
  proposition:  \\
	
  \noindent\textbf{Claim II:} If $X\to T$ has no generic
  indistinguishable points then neither does $XX$. 
	
  \proof Suppose that $P,Q$ are generic $XX$-indistinguishable
  points. Then for all generic $r\in M$ there is some $s$ such that
  $(p_1,r)$ is $X$-indistinguishable from $(p_1,s)$. Since there are
  no generic $X$-indistinguishable points, we get that -- fixing a
  generic $r$ -- $(p_1,r)=(q_1,s)$. In particular $p_1=q_1$. So we are
  reduced to showing that there are no $XX$-indistinguishable points
  of the form $(p,q)$ $(p,r)$. Fix $t$ generic over all the data and
  let $\{p_1,\dots, p_k\}:=X_t(p)$. If $(p,q)$ and $(p,r)$ are
  $XX$-indistinguishable then for generic $s\in T^{(q,p_1)}$ there is
  some $p_i$ such that $s\in T^{(r,p_i)}$. Since $T^Q$ is
  $\Mm$-stationary for generic $Q$ this $i$ cannot depend on $s$. So
  $(q,p_1)$ is $X$-indistinguishable from $(r,p_i)$. So $q=r$. This
  shows that there are no generic
  $XX$-indistinguishable points. \qed$_\text{Claim II}$\\
	
  Now we can construct the desired equivalence relation: \\
	
  \noindent\textbf{Claim III:} Define $q\sim p$ if for every
  generic $r$ there exists $s$ such that $(p,r)$ is
  $X$-indistinguishable from $(q,s)$. Then $\nsim$ is an equivalence
  relation once we remove finitely many points.  \proof There are
  finitely many points $P\in M^2$ such that $T^P$ is generic in
  $T$. Removing their coordinates, we may assume those do not
  exist. So $\nsim$ is generically symmetric, and therefore symmetric
  on co-finitely many points. So we only have to check
  transitivity. Assume that $p\sim q\sim r$. Then for every generic
  $r$ there exists $s$ as in the claim. Our assumptions imply that $r$
  is generic over $q$, so there is $u$ such that $(q,s)\sim (r,u)$ and
  since $(p,r)\sim (q,s)$ and by genericity $T^{(p,r)}$, $T^{(q,s)}$
  and $T^{(r,u)}$ are   irreducible, the claim
  follows. \qed$_\text{Claim III}$ \\ 

  By construction of $\nsim$ and by Claim I, if $([p], [q])$ and
  $([r], [s])$ are $(X/\nsim)( X/\nsim)$-indistinguishable (in
  $M^2/\nsim$) then $[p]=[r]$ and $[q]=[s]$. This finishes the proof
  of the proposition.
\end{proof}

The last proposition shows that, at the cost of passing, to $M/\nsim$
(and replacing $X/\nsim$ with $(X/\nsim)\circ (X/\nsim)^{-1}$) we may
assume that there are no $X$-indistinguishable points. Indeed, by (the
proof of) Lemma \ref{big-generic-family} there are $\Mm$-definable
families of plane curves of arbitrary dimension, satisfying the
conclusion of Theorem \ref{cleanup}, and admitting no
indistinguishable points.  

\begin{rem}
  The structure $(\Mm/\nsim, X/\nsim)$ is interpretable in $\Mm$, and
  therefore already in our original algebraically closed field. There
  is, therefore, no harm assuming that $(\Mm/\nsim, X/\nsim)$ is, in
  fact, $\Mm$.
\end{rem}

\subsection{Counting geometric intersection points}
Intuitively, if $s\in T$ is generic, $P\in X_s^0$ and $t\in T^P$ then
we expect $\#(X_t^1\cap X_s^1)$ non-generic -- because $P$ is a
superfluous intersection point. If that were the case $P$ would be
member of the $\Mm$-$s$-definable algebraic set of all those points on
$X_s$ such that $\#(X_t^1\cap X_s^1)$ is non-generic for all generic
$t$ passing through them. In the present sub-section we show that this
intuition is as true as one could expect in the present
setting. Namely, that the above intuition holds on the level of
$\bar X^1$, namely on the level of closures of the $1$-dimensional
components of members of $X$.

First we study the intersection of any two field generic curves. Our
first observation is obvious, but we isolate it for future reference:
\begin{lemma}\label{generic-int}
  Let $A\subseteq M$ be small and $P$ any point, generic enough over
  $\0$. Then for any $t\in T^{P}$ field generic over $A$ and any
  curve, $C$, field-definable over $A$ we have that
  $\acl_\M(A)\cap C\cap X_t\subseteq \{P\}$.
\end{lemma}
\begin{proof}
  The assumption that $P$ is generic enough in $M^2$ assures that
  $T^P$ is $\Mm$-irreducible, and since $\dim(T)\ge 2$ it is also
  infinite. Thus, if some field generic $t\in T$ satisfies
  $Q\in X_t\cap C\cap \acl_\M(A)$ then the set
  $\{t'\in T^P: Q\in X_{t'}\}$ is $\Mm$-definable of maximal rank,
  implying that $Q$ is $X$-indistinguishable from $P$. So $Q=P$.
\end{proof}

\begin{lemma}
  Let $X\to T$ be as above. Let $t,u\in T$ be field generic
  independent over $\0$ (satisfying the same field-type over
  $\0$). Then  
  \[
    X_t \cap X_u  =  \wbar{X^1_t} \cap \wbar{X^1_{u}} \wo C
  \]
  Where $C$ is the set of frontier $P$-indistinguishable points for
  some (equivalently, any) $P\in X_t\cap X_u$.   
\end{lemma}

\begin{proof}
  Let $P\in X_t\cap X_u$ be any point. By the previous lemma $P$ is
  field generic in $X_t$ and by symmetry $P$ is also generic in
  $X_u$. Since $P$ was arbitrary
  $X_t\cap X_s\subseteq \wbar{X^1_t} \cap \wbar{X^1_{u}}$, with the
  desired conclusion. More precisely,
  \[2\dim(T)=\dim(P,t,u)=\dim(u)+\dim(P/u)+\dim(t/P,u)=\dim(T)+1+\dim(t/P,u)\] 
  implying that $u,t$ are independent generics in $T^P$, so that -- by
  definition of frontier indistinguishable points -- $C\cap X_t\cap
  X_s=\0$.  
\end{proof}	


In the application our main concern will be in the situation where $s\in T$  is generic but $X_u\in T^P$ for
some $P\in X_s^0$. We prove: 

\begin{lemma}\label{cleanup-prep}
  Let $s\in T$ be generic, $P\in X_s^0$ and $(t,u)\in T^P\times T^P$
  generic independent from $s$ over $P$. Then either $P\in
  \acl_\Mm(s)$ or $\#(\wbar{X^1_t} \cap
  \wbar{X^1_{s}})<\#(\wbar{X^1_t} \cap \wbar{X^1_{u}})$.  
\end{lemma}

\begin{proof}
  We assume that $P\notin \acl_\Mm(s)$ (as we will see later on, this
  assumption will ultimately lead to a contradiction). So $P\in X_s$
  is $\Mm$-generic. \\
	
  \noindent\textbf{Claim I}:  $\tp_\Mm(s,t)=\tp_\Mm(t,u)$. 

  \proof By assumption $T$ is $\Mm$-stationary. By the choice
  of $X$ the point $P$ is $\M$-generic (over $\0$) and therefore
  $T^P$ has a unique $\Mm$-generic type $p$. Thus $T^P\times T^P$ has
  a unique $\Mm$-generic type, denoted $p\otimes p$ and by
  construction $(t,u)\models p\otimes p$. It will now suffice to show
  that $(s,t)\models p\otimes p$. Indeed, as $t$ is generic in $T^P$
  over $s$ this reduces to proving that $s\models p$. Our assumption
  that $P\notin \acl_\Mm(s)$ implies that
  $\dim_\Mm(P,s)=\dim_\Mm(s)+1$. So
  $\dim_\Mm(s/P)=\dim_\Mm(T)-1=\dim_\Mm(T^P)$. As $p$ is the unique
  type in $T^P$ of maximal dimension, the claim is proved.  

  \qed$_{\text{Claim I}}$\\ 
	
  It follows that $\#(X_t\cap X_s)=\#(X_t\cap X_u)$.  By Lemma
  \ref{generic-int} $X_s^0\cap X_t=\{P\}$.  For exactly the same
  reason $\Fr(X_s)\cap X_t=\0$. \\
%  contains only 
%  $P$-indistinguishable points. But as $s\in T^P$ is $\Mm$-generic (as
%  follows from Claim I) all $P$-indistinguishable points are, in fact,
%  contained in $X_s$, so cannot be in $\Fr(X_s)$. Thus, $\Fr(X_s)\cap
%  X_t=\0$. \\ 
	
  \noindent\textbf{Claim II}: Let $Q\in X_s\cap \acl_\M(t)$  then $Q\in
  \acl_\M(s)$. 
	
  \proof First, observe that as $\dim(T)>2$ and $P$ is field generic
  over $\0$ we get  	
  \[\dim\{s'\in S: P\in X_{s'}^0\}>0,\] 
  i.e., $s\notin \acl_\M(P)$. Let
  $\phi(x,t)$ isolate $\tp_\M(Q/t)$. By compactness we may assume that
  for any $t'\equiv^\M_P t$ the formula
  $\phi(x,t')$ is algebraic. Consider  
  \[F:=\{Q': \dim\{t\in T^P: Q'\models \phi(x,t)\}\ge \dim(T)-2\}.\]
  Then $\dim(F)\le 1$ and $Q\in F$.  Therefore, as $F$ is definable
  over $P$, we get that $F\cap X_s$ is finite, so $Q\in
  \acl(s)$. \qed$_\text{Claim II}$ \\ 
	
  It follows from the above claim that $X_s\cap X_t^0=\0$. Indeed, the
  claim implies that any $Q\in X_s\cap X_t^0$ is algebraic over $s$,
  now apply Lemma \ref{generic-int}.  
%  and therefore $X$-indistinguishable . But the previous lemma all
%  $P$-indistinguishable points are generic in $X_t$, so in particular
%  not in $X_t^0$.
	
  Using the same claim again we see that if $Q\in X_s\cap \Fr(X_t)$
  then $Q$ is frontier $P$-indistinguishable. So we get that
  $B:=X_s\cap \Fr(X_t)\subseteq C$, where $C$ is the set of all
  frontier $P$-indistinguishable points (in the notation of the
  previous lemma).
		
  Summing up all of the above, together with the previous lemma, we get: 
  \[ \#(X_t\cap X_s)=\#(X_t\cap X_u).\]
  But 
  \[\#(X_t\cap X_s)=\#(\wbar{X_t^1}\cap \wbar{X_s^1})-|B|+1\] where
  $P$ accounts for the extra point on the left hand side. On the other
  hand
  \[\#(X_t\cap X_u)=\#(\wbar{X_t^1}\cap \wbar{X_u^1})-|C|\] 
  and as $B\subseteq C$ the desired conclusion follows. 	
\end{proof}

The previous lemma gives us the advantage of working with families of
closed curves, allowing us to use intersection theory. The fact that
the family we will be working with is (a priori) only definable in the
full structure, and not necessarily in $\Mm$ will not be of
importance, as we will show that the conclusion of the previous lemma
leads to a contradiction, unless for generic $s\in T$ and $P\in X_s^0$
we have that $P\in \acl_\Mm(s)$.



\subsection{Multiplicities}

We remind that if $X, Y \subset M^2$ are curves and $Q\in X\cap Y$ is
a point on both, then the intersection multiplicity of $X$ and $Y$ at
$Q$ is defined as
\[
\mult(X,Y; Q) = \dim_k \Oo_{M^2,Q} / I_X I_Y
\]
where $I_X$ and $I_Y$ are the ideals cutting out the germs of $X$ and
$Y$ around $Q$. In the application the curves $X$ and $Y$ will be
generic members of definable families of curves, say $X=X_a$ and
$Y=Y_b$. Viewing them as curves over the fields $k(a)$ and $k(b)$ (the
respective function fields of $\locus(a)$ and $\locus(b)$) the
intersection multiplicity of $X_a, Y_b$ at a point $Q\in \acl(a,b)$ is
given by
$$
\mult(X_a,Y_b; Q) = \dim_{k(a,b)^{alg}} \Oo_{M^2 \otimes
  k(a,b)^{alg}, Q} / I_{X_a} I_{Y_b}
$$
where $I_{X_a}, I_{Y_b}$ are the ideals cutting out the germs at $Q$
of $X_a \otimes k(a,b)^{alg}$ and of $Y_b \otimes k(a,b)^{alg}$
respectively, and $k(a,b)$ is the function field of $\locus(a,b)$.

The key local property of the intersection multiplicity used below is
given by the following lemma:

\begin{lemma}
  \label{mult-ineq}
  Let $R$ be a regular local ring over a field $k$, and 
  $I_1$, $I_2$, $I_3$ ideals such that $R/I_1$, $R/I_2$, $R/I_3$ are
  regular. Assume that $R/(I_1 I_2)$,$R/(I_2I_3)$ and $R/(I_1I_3)$ are
  finite-dimensional $k$-vector spaces. Then
  $$
  \dim_k R/(I_1I_2) \geq \min \{ \dim_k R/(I_1I_3), \dim_k R/(I_2I_3)
  \}
  $$
\end{lemma}

\begin{proof}
  By symmetry of the statement it suffices to show that if
  $$
  \begin{array}{l}
    \dim_k R/(I_1I_2) \geq \dim_k R/(I_1I_3), \\
    \dim_k R/(I_1I_2) \geq \dim_k R/(I_2I_3), \\
  \end{array}
  $$
  then $\dim_k R/(I_1I_3) = \dim_k  R/(I_2I_3)$.
	
  By regularity of $R/I_2$, all 0-dimensional quotient algebras are of
  the form $k[a]/(a^n)$ for some generator; for two such algebras
  $k[a]/(a^n), k[a]/(a^m), n > m$ there exists a natural reduction
  morphism $k[a]/(a^n) \to k[a]/(a^m)$. It follows from the first
  inequality above that there exists a morphism of this form
  \[f: R/(I_1I_2) \to R/(I_3I_2) \cong R/I_3 \otimes R/I_2.\] 
  We get the following diagram:
  $$
  \xymatrixcolsep{4pc}\xymatrix{
    R/I_2 \ar[r] \ar[d]^p  & R/I_3 \otimes R/I_2 \ar@<.5ex>[d]^{\id
      \otimes p} \\
    R/(I_1I_2) \ar[r]_-{i_{12}} \ar[ur]^f & R/I_3 \otimes R/(I_1I_2)
    \ar@<.5ex>[u]^{h}\\
  }
  $$
  where $h$ is defined by $a \otimes b \mapsto a \cdot f(b)$.
	
  One observes that both morphisms $h$ and $\id \otimes p$ are
  surjective, and since $R/I_3 \otimes R/I_2$ and
  $R/I_3 \otimes R/(I_1 I_2)$ are finite-dimensional vector spaces,
  they are bijective and so isomorphisms.  By a similar argument,
  $R/I_3 \otimes R/(I_1I_2)$ is isomorphic to $R/I_3 \otimes R/I_1$,
  and therefore $R/I_3 \otimes R/I_1$ is isomorphic to
  $R/I_3 \otimes R/I_2$.
\end{proof}

Geometrically, the above lemma expresses the fact that if $X,Y$ and
$Z$ are curves in $M^2$ all meeting at a common point, $Q$, regular on
all three, and if $\mult(X,Y;Q)=\mult(X,Z;Q)$ then $\mult(Y,Z;Q)\ge
\mult(X,Y;Q)$. We point out that this lemma is not true if regularity
of all curves in question at $Q$ is not assumed (e.g., if $Q$ is a branch
point of $X$, but is regular on $Y$ and $Z$). 

\begin{lemma}\label{invariant-mult}
  Let $s\in T$ be generic, $P \in X^0_s$ and $t\in T^P$ generic over
  $s$. Let $Q \in X_t \cap X_s$.  Assume that all geometric
  intersection points (that is, defined over $\acl(t,s)$) are
  regular. Then there exists a number $m$ such that
  $\mult(X_{t'}, X_s, Q')=m$ whenever $t',Q'$ are such that
  $\tp(t,s,Q) = \tp(t',s,Q')$.
\end{lemma}

\begin{proof}
  In algebro-geometric terms, to the type $\tp(t,s,Q)$ corresponds a
  point $W \in X_t \otimes k(t,s) \cap X_s \otimes k(t,s)$ which is
  regular on both $X_t$ and $X_s$, with a residue field which is an
  algebraic extension of $k(t,s)$. Let $Z$ be the piece of the fiber
  product $X_t \times_{M^2} X_s = \Spec \Oo_{M^2,W}/I_1 I_2$ supported
  at $W$, and let $Z = \Spec A$. If both $X_t$ and $X_s$ are regular
  at $W$, then $\widehat{\Oo_{M^2,W}/I_1} \cong K[[\epsilon]]$ where
  $K$ is the residue field of $\Oo_{M^2,W}/I_1$. Further, $A$ is
  isomorphic to a proper quotient of $\widehat{\Oo_{M^2,W}/I_1}$ and
  so is of the form $K[\epsilon]/(\epsilon^{m+1})$.

  To a point of intersection $Q'$ of $X_t$ and $X_s$ such that
  $\tp(t,s,Q)=\tp(t,s,Q')$ corresponds one of the irreducible
  components of $\Spec A \otimes k(t,s)^{alg}$, and multiplicity is
  equal to the dimension of its coordinate algebra over
  $k(t,s)^{alg}$. It is clear that this does not depend on the
  irreducible component, and equals $m$.
\end{proof}

We can now show: 
\begin{lemma}\label{cleanup-equality}
  Let $s\in T$ be generic, $P\in X_s^0$. Let $Q\in X_s$ be generic
  over $P$ and $t,u\in T^P\cap T^Q$ independent generics. Assume that
  $\tp_\M(t/P,Q)=\tp_\M(u/P,Q)$ and $\tp_\M(s)=\tp_\M(t)$. Then
  $\mult(X_t, X_s,Q)\le \mult(X_u,X_t,Q)$.
\end{lemma}

\begin{proof}
  By our choice of $X\to T$ we know that $\dim(P)=2$ and as
  $\dim(T) > 2$ we get $\dim(T^P)\ge 2$. Moreover, as in the proof of
  Claim II of Lemma \ref{cleanup-prep}, $s\notin \acl(P)$. This
  implies that $Q\ind_\0 P$, whence
  $\dim(T^P\cap T^Q)=\dim(T)-2\ge 1$. Thus, if $t\in T^P\cap T^Q$ is
  generic we have
  \[\dim(T)+2=\dim(t,P,Q)=\dim(Q/t,P)+\dim(t/P)+\dim(P).\]
  Since $\dim(Q/t,P)=1$ this implies that $\dim(t/P)=\dim(T)-1$, so $t$ is generic in $T^P$. 
  Similarly, if $t,u\in T^P\cap T^Q$ we have: 
  \[2\dim(T)=\dim(P)+\dim(t,u/P)+\dim(Q/t,u,P) \] 
  and as $Q\in \acl(t,u,P)$ this implies that
  $\dim(t,u/P)=2\dim(T)-2$, i.e., $(t,u)$ are independent generics in
  $T^P$. Since $P$ is generic over $\0$ this implies that $t,u$ are
  independent generic over $\0$ as well.  
	
  Thus $Q$ is regular on both $X_u$ and $X_t$. Assume towards a
  contradiction that $\mult(X_s,X_t,Q)>\mult(X_t,X_u,Q)$. By
  Lemma~\ref{invariant-mult}, since $\tp_\M(t/P,Q) = \tp_\M(u/P,Q)$, also
  $\mult(X_s,X_u,Q)=\mult(X_s,X_t,Q)$, and so
  $\mult(X_u,X_s,Q)>\mult(X_t,X_u,Q)$. But by Lemma~\ref{mult-ineq},
  $\mult(X_t,X_u,Q) \geq \mult(X_u,X_s,Q)$, a
  contradiction.
\end{proof}

The global implication of the previous (local) lemma is: 
\begin{lemma}
  Let $s\in T$ be generic $P\in X_s^0$. Let $t,u\in T^P$ be
  $\M$-independent generic over all the data (satisfying the same $\M$-type). Then 
  \[
    \sum_{Q\in \wbar{X_s^1}\cap \wbar{X_t^1}} \mult({\wbar{X_s^1},
      \wbar{X_t^1}, Q})< \sum_{Q\in \wbar{X_u^1}\cap \wbar{X_t^1}}
    \mult({\wbar{X_u^1}, \wbar{X_t^1}, Q}) 
  \]
\end{lemma}

\begin{proof}
  For simplicity we will assume that all curves in question have a
  unique 1-dimensional component (in the sense of $\M$). Otherwise we repeat the argument component by component.
	
  Denote $\mathcal Q:=\wbar{X_s^1}\cap \wbar{X_t^1}$. By Lemma
  \ref{generic-int} all points in $\mathcal Q$ are $\M$-generic in
  both $X_s$ and $X_t$.
%  and $\mathcal
%  Q_1$ the subset consisting of the points generic in $\bar
%  {X_s^1}$. Observe that if $Q\in \mathcal Q\setminus \mathcal Q_1$
%  then $Q\in \wbar{X_{t'}^1}$ for all $\tp(u/P,s)=\tp(t/P,s)$, in
%  particular, $\mathcal Q\setminus \mathcal Q_1$ does not depend on
%  the choice of $t$ (only on its field-theoretic type). So by Lemma
%  \ref{mult-ineq} we get  
%  \[
%    \mult({\wbar{X_s^1}, \wbar{X_t^1}, Q}) \le \mult({\wbar{X_u^1}, \wbar{X_t^1}, Q}).
%  \]
%  Thus, 
%  \[
%    \sum_{Q\in \mathcal Q\setminus \mathcal Q_1} \mult({\wbar{X_s^1},
%      \wbar{X_t^1}, Q})\le  \sum_{Q\in \mathcal Q\setminus \mathcal
%      Q_1} \mult({\wbar{X_u^1}, \wbar{X_t^1}, Q}).
%  \]
  As we assumed that $X_s$ and $X_t$ are
  irreducible, and -- restricting to an irreducible component of
  $T^P$ -- we get by Lemma \ref{invariant-mult} that there exists a
  number $m$ such that whenever $t\in T^P$ is $\M$-generic (of a
  fixed type)  
  $\mult(\wbar{X_s^1},\wbar{X_t^1}, Q)=m$ for any $Q\in
  \wbar{X_s^1}\cap \wbar{X_t^1}$. Similarly, there
  exists $n$ such that whenever $t,u\in T^P$ are $\M$-independent generics,
  satisfying  the same $\M$-type
  $\mult(\wbar{X_u^1},\wbar{X_t^1}, Q)=n$ for any $Q\in
  \wbar{X_s^1}\cap \wbar{X_t^1}$.
	
  By Lemma \ref{cleanup-equality} we get $m\le n$. By Lemma
  \ref{cleanup-prep} we have that
  $\#\mathcal Q<\#(\wbar{X_u^1}\cap \wbar{X_t^1})$, with the desired
  conclusion.
  
%  which is precisely what we showed  But by Lemma \ref{cleanup-prep} we
%  know that 
%  \[\#(\wbar{X_s^1}\cap \wbar{X_t^1})<\#(\wbar{X_u^1}\cap
%    \wbar{X_t^1})\] 
%  and as we have just shown that the two sets share
%  the same set $\mathcal Q\setminus \mathcal Q_1$ of non-generic
%  points, the desired conclusion follows.
\end{proof}


We point out that the last lemma is not true if our family has
indistinguishable points: If, in the notation of the previous lemma,
$P$ is $X$-indistinguishable from a point $Q$ which is a branch point
of $X_s$. This is the only place where assuring that $X$ has no
indistinguishable generic points was crucial for the proof.


\begin{proof}[Proof of Proposition \ref{cleanup}]
  Fix a family $X\to T$ as provided by Corollary
  \ref{cleanup-assumptions}. Quotienting by an equivalence relation
  with finite classes, we may assume, by Proposition
  \ref{cleanup-ind}, that there are no $X$-indistinguishable generic
  points.  If for $s\in T$ generic $X_s^0\subseteq \acl_\Mm(s)$ then,
  by compactness we have nothing to prove. So assume that this is not
  the case. We will derive a contradiction.

  %We first show we may assume $M$ to be regular and proper. Indeed, 
  
  We have already identified $M$ with its regular locus (by simply
  removing all singularities). So we may assume $M$ is regular. Now
  choose a proper curve $M_1$ containing $M$ as an open
  subset. Consider the normalization $M_2$ of $M_1$. Note that $M_1$
  projects onto $M_1$ and is regular since normal varieties have
  singularities in codimension 2.  The projection is an isomorphism
  after restriction to the regular locus of $M_1$: local rings of
  regular points of $M_2$ are integral closures of local rings of
  regular points of $M_1$, and regular local rings of dimension 1 are
  integrally closed. It follows that there exists an open embedding of
  $M_1$ into $M_2$, and hence of $M$ into $M_2$.
  
%  Consider the first order structure $M^*$ with the universe $M_2$
%  which contains the restriction of the family $X$ to $M_0$,
%  interpreted as a subset of $(M_2)^l$ for some $l$ via the embedding
%  $M_0 \hookrightarrow M_2$ that has been just picked. Clearly, $M^*$
%  is non-locally modular. Note that any $M^*$-definable subset of any
%  Cartensian power $(M_0)^l$ is definable in the original structure
%  $\mathcal M$, and vice versa, by stable embeddedness.

  Let $s\in T$ be field generic and $P\in X_s^0$ such that
  $P\notin \acl_\Mm(s)$, $Q\in S$ a field generic point. Let
  $t,u\in T^P\cap T^Q$ be field independent generic over all the
  data. By our assumption it follows from Lemma \ref{cleanup-prep}
  that
  $\#(\wbar{X_s^1}\cap \wbar{X_t^1})<\# (\wbar{X_u^1}\cap
  \wbar{X_t^1})$, where the closure is taken in (some Cartesian power
  of) $M_2$.

  As follows from intersection theory on proper regular varieties, the
  intersection number is stable in algebraic families (\cite{fulton},
  Corollary~10.2.2). Therefore the sum of local intersection
  multiplicities over all intersection points should be the same for
  pairs $X_t, X_s$ and $X_t, X_u$. This is in direct contradiction
  with the previous lemma.
\end{proof}

Though Proposition \ref{cleanup} suffices for our needs in the current
paper, we give the proof of Theorem \ref{cleanup-general}, whose
statement is cleaner, and may be of interest on its own right.

\begin{proof}[Proof of Theorem \ref{cleanup-general}]
  Let $S$ be any $\Mm$-definable curve. Absorbing the parameters
  required to define $S$, we may assume that $S$ is
  $\0$-definable. Let $X\to T$ be any $\Mm$-definable family of plane
  curves satisfying the assumptions of Proposition \ref{cleanup}. For
  simplicity, we may also assume that $X$ satisfies the conclusion of
  the proposition. Consider the family $S\circ X\to T$. Our assumption
  implies that for a generic $t\in T$ the only isolated points of
  $S\circ X_t$ are of the form $S^0\circ \wbar {X_t^1}$ (as
  $\wbar {X_t^1}=\wbar {X_t}$). Applying Proposition \ref{cleanup} to
  $S\circ X_t$ (for some generic $t\in T$) we get a curve
  $Z_t\sim S\circ X_t$ such that $Z_t^0=\0$ (and $Z_t$ is definable
  over $t$). So
  $S^0\subseteq \{P\in S: P\circ X_t\setminus Z_t\neq \0\}$. Note that
  the right hand side is $t$-definable (and finite). So
  \[
    S^0:=\bigcap_{t\in T \text{ generic}} \{ P\circ X_t\setminus Z_t\neq \0\}
  \]
  and by definabilty of Morley rank the right hand side is
  $\0$-$\Mm$-definable.  
\end{proof}

Note that the proof of Theorem \ref{cleanup-general} follows almost
formally from Proposition \ref{cleanup}, and has little to do with the
topological definition of the set of $0$-dimensional components. The
only property of $0$-dimensional components used in the proof is that
if $D$ has no $0$-dimensional components then
$(S\circ D)^0\subseteq S^0\circ D$.

Also the proof of Proposition \ref{cleanup} does not seem endemic to
algebraic geometry. The only algebro-geometric ingredients used in the
proof are
\begin{enumerate}
\item Finiteness of the frontier of (plane) curves. 
\item Continuity of the intersection number in algebraic families of plane curves in proper regular surfaces. 
\item The multiplicity inequality of Lemma \ref{mult-ineq}
\end{enumerate} 

The last two of these three properties seem to have satisfactory
analogues in a variety of analytic and topological settings such as
the invariance of the topological degree under homotopy (and see,
e.g., \cite[Lemma 4.19, Lemma 4.20]{HaKo}). The multiplicity
inequality is a refinement of ideology that tangency should be an
equivalence relation. In many respect this is the cornerstone upon
which Zilber's Trichotomy -- suggesting the construction of a field
from purely geometric, even combinatorial, data -- relies. It is
therefore reasonable to expect to have natural analogues in any
context in which one can reasonably hope to prove this trichotomy.
	  
\section{Concluding remarks}
We can finally prove the main theorem of the present paper: 	  
\begin{thm}[Main theorem]
  \label{main-theorem}
  Let $M$ be an algebraic variety of dimension 1 defined over an
  algebraically closed field $k$. Let $X \subset T \times M^2$ be a
  family of 1-dimensional subsets of $M^2$. Then the structure
  $\mathcal{M}=(M,X)$ interprets an infinite field $k$-definably
  isomorphic to $k$.
\end{thm}

\begin{proof}
  By Lemma \ref{cleanup-prep} and Proposition ~\ref{cleanup} there exists a two-dimensional 
  family of curves definable in $(M,X)$ with generic fibre of pure
  dimension 1. By Theorem~\ref{pure-dim-first-group} 
  a one-dimensional group $G$ is definable in $\mathcal{M}$. The structure
  induced on $G$ by $\mathcal{M}$ is non-locally modular, so the theorem is
  reduced to the case when there is a group structure on $M$. 

  A non-locally modular strongly minimal group $G$ has a definable set
  $Z \subset G^2$ which is not a coset.  Use it to produce the
  definition of a field by Theorem~\ref{pure-dim-field}. That the
  field is pure and definably isomorphic to $k$ follows from
  \cite[Theorem 4.15]{poizat}.
\end{proof}

As an immediate application we get: 
\begin{cor}
  Let $\Mm$ be $\aleph_0$-stable. If $\Mm$ admits a rank-preserving
  interpretation in an algebraically closed field, $K$, then either
  $\Mm$ is 1-based or a field $K$-algebraically isomorphic to $K$ is
  interpretable in $\Mm$.
\end{cor}
\begin{proof}
  Because $\Mm$ is interpretable in an algebraically closed field, it
  is of finite Morley rank. So $\Mm$ is 1-based if and only if every
  strongly minimal set interpretable in $\Mm$ is locally
  modular. Thus, if $\Mm$ is not 1-based it interprets a non-locally
  modular strongly minimal set. Since the interpretation of $\Mm$ in
  $K$ is rank preserving, the universe of this strongly minimal set is
  the set of $K$-rational points of some curve $\mathcal C$ definable
  over $K$. Now apply the above theorem for $\mathcal C$.
\end{proof}

Consider, for example, $\mathcal C$ an algebraic curve (of genus at
least 2), $\mathcal J(\mathcal C)$ its Jacobian, and
$\mathcal J:=(J,+,C)$ the pure group structure on
$\mathcal J(\mathcal C)$ expanded by a unary predicate for the image
of $C$ in $\mathcal J(\mathcal C)$ under the Abel-Jacobi map.  Then
$\mathcal J$ admits a rank preserving interpretation in any
algebraically closed field $K$ over which $\mathcal C$ is defined. It
is not locally modular because the image of $\mathcal C$ in
$\mathcal J(\mathcal C)$ does not coincide with a any coset of a
subgroup of $(J,+)$. This slightly simplifies the main ingredient in
Zilber's proof of \cite{ztorelli} -- the interpretation of $K$ in
$\mathcal J$.

Similarly, if $\mathfrak G$ is the full Zariski structure on a simple
algebraic group then its reduct $\mathcal G$ -- the pure group
structure -- is clearly interpretable in any algebraically closed
field. It is standard to verify that this interpretation is
(necessarily) rank preserving. This gives a new proof of the main
result of \cite{Poizat-ACFs}, Poizat's model theoretic (baby) version
of Borel-Tits, stating that any group isomorphism between two simple
algebraic groups over algebraically closed fields can be decomposed
into an isomorphism of the fields followed by a quasi-isomorphism of
the algebraic varieties. This new proof does not make any use of the
analysis of bad groups.

The statement of the above corollary omitting the requirement that the
interpretation be rank preserving is equivalent to the full restricted
trichotomy conjecture for algebraically closed fields. We hope that
the tools developed in the present paper could help settle this long
standing open problem.

\bibliography{reducts}

\newpage


\noindent {\sc Department of Mathematics \\
Ben Gurion University of the Negev \\
P.O.B. 653 \\
Be'er Sheva 84105 \\
Israel\\}
{\tt hassonas@math.bgu.ac.il}
\vspace{2ex}

\noindent {\sc Laboratoire Paul Painlev\'{e}\\
 CNRS U.M.R. 8524 \\
59 655 Villeneuve d'Ascq Cedex​ \\
France\\}
{\tt Dmitry.Sustretov@math.univ-lille1.fr\\}


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reducts.bbl
\begin{thebibliography}{41}
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\providecommand{\url}[1]{\texttt{#1}}
\expandafter\ifx\csname urlstyle\endcsname\relax
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\end{thebibliography}
