On K3 Correspondences
Madeeha Khalid
We consider relationships between families of K3 surfaces, in the context of string theory. An important ingredient of string theory also of interest in algebraic geometry is T-duality. Donagi and Pantev \cite{DP} have extended the original duality on genus one fibred K3 surfaces with a section to the case of any genus one fibration, via a Fourier-Mukai transform. We investigate possibilities of extending this result to the more general case of non fibred K3s, in the context of C\u ald\u araru's conjecture \cite{C1}. We also provide an explicit construction of a quasi-universal sheaf associated to the classical Mukai correspondence. (see \cite{M1}).
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\begin{document}

\title{On K3 Correspondences}
\author{Madeeha Khalid}



\date{}

\maketitle


\centerline{\it{Dedicated to Ammi, Abbu, Myda, Baji and Qudrat-Ullah Shahab}}
\centerline{\it{with gratitude and love}}



\begin{abstract}





We consider relationships between families of K3 surfaces,
in the context of string theory.
An important ingredient of string theory also of interest in algebraic
geometry is T-duality.
Donagi and Pantev \cite{DP} have extended the original duality on genus one
fibred K3 surfaces with a section to the case of any genus one fibration,
via a Fourier-Mukai transform.  
We investigate possibilities of
extending this result to
the more general case of non fibred K3s, in the context of 
C\u ald\u araru's conjecture \cite{C1}.

We also provide an explicit construction of a quasi-universal sheaf
associated to the classical Mukai correspondence. (see \cite{M1}).
\end{abstract}



\section {Context and Background}


\paragraph{Calabi--Yau manifolds}
A class of geometric spaces of interest to both mathematicians and 
physicists are Calabi--Yau (CY) manifolds.

\begin{defn}
A CY manifold is a complex manifold $X$ with a non vanishing top
dimensional global holomorphic form and with $H^i(X,\Oh) = 0$ for $0<i<n$,
where $n=\dim_\C X$.

Note: Some definitions in the literature omit the second condition.
\end{defn}

Two dimensional CY manifolds are called K3 surfaces.
There are dualities between different string theories that induce
highly nontrivial relations amongst the geometric structures of the
associated CY manifolds. One such duality is mirror symmetry, to which
Candelas et al \cite{Cetal}, Strominger, Yau and Zaslow \cite{SYZ}
%Vafa and Witten \cite{VW}% 
and, most recently, Gross and Seibert
\cite{GS} have made major contributions.

%\paragraph{Derived Categories}
%Topological string theory suggests that the derived category of coherent
%sheaves on a CY manifold is an important invariant of the corresponding
%string theory (work of Aspinwall and Donagi \cite{AD} and Douglas \cite{D}).
%However, the machinery involved is rather laborious, motivating a quest for
%simpler geometric criteria for comparing the different physical theories.
%In some string theory, one has non-trivial two forms with values in 
%$U(1)$, called B-fields.  These fields correspond to a generalisation
%of a $U(1)$ guage connection. In the cases that the B-field takes values
%in a finite subgroup of $U(1)$ it can be viewed as an element of
%$\h^2(X, \Oh^*)_{tors}$.
Lately there has also been much interest in the derived category
of twisted sheaves on CYs and dualities between them. We give the
definition of a 'twisted sheaf.'

\begin{defn}
Given $\theta \in \h^2(X, \Oh^*)$ and $\{ U_i \}$ a 
sufficiently fine  
open cover of $X$,
let ${\theta}_{ijk}$ be a Cech cocycle representative of $\theta$.
Then a $\theta$-twisted sheaf on $X$ is a collection
$\{ F_i \} , \{ {\phi}_{ij} \}$ where $F_i$ is a sheaf on $U_i$  and
${\phi}_{ij}$ are isomorphisms relating $F_i$ and $F_j$ on $U_i \cap U_j$,
such that
$ {\phi}_{ij} {\phi}_{jk} {\phi}_{ki} = {\theta}_{ijk}$ on
$U_i \cap U_j \cap U_k$. 
\end{defn}


%Another motivation for studying twisted sheaves is due to the
%appearance of orbifold CYs in string theory. This suggests moving away from
%conventional Euclidean space towards a noncommutative one. The notion of
%a noncommutative variety is still not well defined in general and is
%currently an area of intense research. However there is one simple
%case of a well defined noncommutative space, and that is a
%$gerbe$ associated to an element of $H^2(X,\Oh^*)$ (where $X$ is
%the underlying manifold). This has a natural interpretation in terms
%of twisted sheaves on $X$. The gerbe $\theta$ associated to $X$ can
%be thought of as a space with the same underlying set of points as $X$
%with the extra condition that a sheaf on the gerbe is a $\theta$-twisted
%sheaf on the underlying space $X$. Since sheaves on the gerbe do not
%satisfy the usual cocycle condition, in a sense the gerbe is a
%``noncommutative space'' (for more details see Hitchin \cite{H1}).

%Then the following
%beautiful conjecture by C\u ald\u araru, though restricted to K3
%surfaces, offers some direction.

C\u ald\u araru has posed a beautiful conjecture relating equivalences
between derived categories of twisted sheaves on K3 surfaces.
Before we state the conjecture we give some facts about K3 surfaces.  
The second cohomology of a K3 surface is generated (over $\Q$) by the
direct sum of its Picard lattice $\Pic(X)$ and its transcendental lattice
$T_X$. The elements of the Picard lattice are the algebraic 
classes (Chern classes of holomorphic vector bundles on X).
The lattice $T_X$
is equal to the orthogonal complement of $\Pic(X)$ in $H^2(X,\Z)$.
The Brauer group $\Br(X)$ of an algebraic K3 surface $X$ is the \'etale 
cohomology group $H^2_{\hbox{\'et}}(X,\Oh^*)$. For an algebraic K3 surface
$X$,
 $$
 \Br(X) \cong T_X ^\vee \tensor \Q/\Z \quad\hbox{(see \cite{C1}).}
 $$

\begin{conj}
Let $X$ and $Y$ be algebraic K3 surfaces with transcendental lattices 
$T_X$ and $T_Y$, and $\al\in\Br X$ and $\be\in\Br Y$. Then the following
are equivalent:
\begin{enumerate}
\item There exists an equivalence $\D^{b}(X,\al)\to\D^{b}(Y,\be)$ of
the derived categories of \hbox{$\al$-twisted} sheaves on $X$ to
$\be$-twisted sheaves on $Y$.
\item There exists a Hodge isometry $\ker\al\to\ker\be$ between the
sublattices of finite index $\ker\al\subset T_X$ and $\ker\be\subset
T_Y$.
\end{enumerate}
\end{conj}


Orlov (see \cite{O})has proved the case when both $\al$ and $\be$ are trivial.
C\u ald\u araru has proved the case for 
certain K3 surfaces where one of $\al$ or $\be$ is non trivial.
His result is a genralisation of Mukai's theorem \cite{M1}.

%Based on a classical correspondence due to Mukai \cite{M1}, 
%C.~Madonna and V.V.~Nikulin \cite{MN},
%have determined elegant number theoretic conditions on $\Pic(X)$ and
%$\Pic(Y)$  such that $X \cong Y$ and the conjecture holds for these.

Most recently Donagi and Pantev \cite{DP} have proved the conjecture for
genus one fibred K3 surfaces.  Their result is a bit more general 
since only one of the K3s has to be algebraic.


Although these provide numerous examples of K3 surfaces for which the 
conjecture is true, to date, in its full generality, the conjecture 
stands unanswered.  

One of the results in my thesis \cite{Kh} which is a step towards an
answer to this question is the following theorem
about Hodge cycles on products of K3 surfaces.

\begin{thm} \label{thm:Fa}
There exists a $19$-dimensional family $\F$ of pairs $(M,Y)$, where $M$ is
a double cover of $\PP^2$, $Y$ is a degree 8 surface in $\PP^5$ and
$(T_M)_{\Q}$ is Hodge isometric to $(T_Y)_{\Q}$.
The generic pairs $(M,Y)$ in $\F$ are non\-fibred K3 surfaces.
\end{thm}

The Hodge isometry between ${T_M}_{\Q}$ and ${T_Y}_{\Q}$ is induced by 
an algebraic cycle 
on the product $Z \in H^{2,2}(M \times Y,\Q)$. This cycle is often referred
to as the {\textbf{isogeny cycle}} or a {\textbf{correspondence}}. 

The main point here is that for a generic pair $(M,Y)$ in the above family,
the isometry induced by the isogeny cycle does not restrict to an embedding
of the integral lattice $T_M$ into $T_Y$ or the other way around. 
Note that in this respect the family above differs from Mukai's family 
of corresponding pairs $(M,Y)$, where $T_Y$ embeds in $T_M$ as a sublattice 
of index 2. 

Based on the idea behind this example we construct infinitely many families
of isogenous K3s.  These families are natural
testing grounds for the conjecture.

In Section 2 we will list some preliminary facts about K3 surfaces
In Section 3 we will explain the Donagi-Pantev result with examples.
In Section 4 we will illustrate the main idea 
with a couple of examples, before giving the general construction.
In Section 5 we will give the explicit construction of the quasi universal
sheaf associated to the classical Mukai correspondence.


\paragraph{Acknowledgments}
I would like to express my deepest thanks to my advisor Ron Donagi,
for his patience and guidance through the span of my graduate
studies.  I would also like to thank Tony Pantev for his generous help
with my questions in algebraic geometry.  
A special thanks is owed to Professor Karen Uhlenbeck and to my family 
and friends for their continued support and encouragement.





\section{K3 Surfaces}

We state some facts about K3 surfaces which we will use later.

\subsection{Topological and analytical invariants}


\begin{notation}
Let $X$ be a compact connected complex surface with the cup product form
$\intf{,}$ on $\h^2(X,\C)$.

Let $L=-E_8\oplus -E_8\oplus H \oplus H \oplus H$ where $E_8$ is the root 
lattice of the exceptional Lie group $E_8$. $H$ is the hyperbolic lattice
with intersection form 
$\left(\begin{array}{rr}0 & 1\\ 1 & 0 \end{array}\right)$.


%%$\left(\array{ 0 & 1 \\ 1 & 0}\right)$.



$L_\C:=L\tensor\C$ with $\intf{,}$ extended $\C$ bilinearly.
For $\omega\in L_\C$, we denote by $[\omega]$ in 
$\PP(L_\C)$ the corresponding line
and set $\Omega=\{[\omega] \in \PP L_\C: \intf{\omega,\omega}=0, 
\intf{\omega, \overline{\omega}} >0\}$.
\end{notation}

\begin{defn}
A K3 surface is a (possibly singular) surface with trivial canonical 
bundle and $b_1(X)=0$.
\end{defn}


\begin{proposition}
Let $X$ be a smooth K3 surface, then $c_1(X)=0$, $c_2(X)=24$ and $\h^2(X,\Z)$
equipped with the cup product pairing is isometric to the lattice $L$. 
\end{proposition}

The Hodge numbers of a K3 surface are completely determined.
\begin{proposition}
Let $X$ be a smooth  K3 surface. 
Then 
$h^{0,1}(X)=h^{1,0}(X)=h^{2,1}(X)=h^{1,2}(X)=0$,
 $h^{0,2}(X)=h^{2,0}(X)=1$, $h^{1,1}(X)=20$.
\end{proposition}
The proofs involve Kodaira-Serre duality and Riemann-Roch for surfaces.
(For more details, see \cite{BW}, Chapter VIII.)

If $X$ and $Y$ are surfaces, an isomorphism of $\Z$-modules
$$\h^2(X,\Z) \ra \h^2(Y,\Z)$$ is called a {\textbf{ Hodge isometry}} if
it preserves the cup product $\intf{,}$,
and its $\C$-linear extension preserves the Hodge decomposition.

We will be interested in the Hodge isometries over $\Q$.


The Neron-Severi group of $X$, denoted by $N_X$, is equal to  
$\Pic(X)/\Pic^0(X)$. In the case $X$ is a smooth K3 surface, $N_X$ is actually 
isomorphic to $\Pic(X)$.
The transcendental lattice of $X$ is $T_X:=N_X^\perp$ in $\h^2(X, \Z)$.


Both $N_X$ and $T_X$ are primitive sublattices of $\h^2(X,\Z)$, but 
are generically 
non-unimodular.

Given two K3 surfaces $X$ and $Y$, a cycle 
$Z\in \h^2(X)_{\Q} \tensor \h^2(Y)_{\Q}$
and projections $p_X, p_Y$ onto the first and second factor respectively, there
exists an associated mapping $f_Z$,
$$
\begin{array}{lcccc}

f_Z & : & \h^2(X,\Q) & \longrightarrow & \h^2(Y,\Q) \\
 & & t & \mapsto & p_{Y*}(p_X^*(t)\cdot Z) 

\end{array}
$$

Sometimes we will denote $f_Z$ by $Z$, for ease of notation.

We use the following definition. 
See \cite{M2}, Definition 1.7.

\begin{defn}
An algebraic cycle $Z$ in $(\h^2(X) \tensor \h^2(Y))_\Q$ is
 an {\textbf{isogeny}}
(or a {\textbf {correspondence}}),
if the homomorphism $Z: \h^2(X,\Q) \ra \h^2(Y,\Q)$ is an isometry.

\end{defn}






Since
$\h^2(X,\Q)=(N_X\oplus T_X)_\Q$ and 
$\h^2(Y,\Q)=(N_Y\oplus T_Y)_\Q$,
$$\h^2(X,\Q)\tensor \h^2(Y, \Q)=[(N_X\tensor N_Y)\oplus (N_X\tensor T_Y)\oplus
(T_X \tensor N_Y) \oplus (T_X \tensor T_Y)]_\Q.$$



The vector space $((N_X\tensor \ty)\oplus
(\tx \tensor N_Y)$ does not contain cohomology classes of type $(2,2)$.

Therefore $Z=Z_N+Z_T$, where $Z_N\in N_X\tensor N_Y$ 
and $Z_T \in T_X\tensor T_Y$.
The $Z_N$ component  is always algebraic, 
hence a Hodge cycle $Z$ is algebraic if and only 
if $Z_T$ is algebraic.


\subsection{Period mapping}
Given a smooth K3 surface $X$, $\h^2(X,\Z)$ is isometric to the lattice $L$.
The choice of an isometry $\varphi: \h^2(X, \Z) \ra L$ determines a 
line in $L_\C$, spanned by the $\varphi_\C$-image of the nowhere vanishing
holomorphic 2-form $\omega_X$.

The identity $\intf{\omega,\omega}=0$ and the inequality 
$\intf{\omega,\overline{\omega}}
>0$ imply that this line considered as a point in $\PP(L_\C)$ belongs
to $\Omega$.  Recall that $\Omega$ is the period domain, 
defined earlier in Section 2.1. 
This point is called the \textbf{period point} of a marked
K3 surface $(X,\varphi)$.

There are two main theorems regarding marked K3 surfaces.

\begin{thm}[Torelli Theorem]
Two K3 surfaces $X$,$Y$ are isomorphic if and only if there exists a Hodge 
isometry $\varphi: \h^2(X,\Z) \ra \h^2(Y,\Z)$.
\end{thm}

\begin{thm}
All points of $\Omega$ occur as period points of marked K3 surfaces.
\end{thm}
(For proofs, see \cite{BW}, Chapter VIII.)


\begin{defn}
A family $p:\X \ra \Delta$ is said to be a \textbf{marked deformation} 
if there exists a trivialization $\varphi: R^2 p_*\Z_\X \ra L$.
\end{defn}

Associated to a marked deformation of K3 surfaces we have 
the period mapping.

\begin{defn}
Suppose we have a marked deformation of K3 surfaces,
the associated \textbf{period mapping} $\tau:\Delta \ra \Omega$ is defined by 
sending $t\in \Delta$ to  the complex line $\varphi_{\C,t}(\h^{2,0}(\X_t))$
in $L_\C$, and then taking the corresponding point in the projective 
space $[ \varphi_{\C,t}(\h^{2,0}(\X_t))  ]\in\PP(L_\C)$. 
\end{defn}

\begin{notation} 
Given a family $p:\X \ra B$ of K3 surfaces, 
let $\Omega_\X$ denote the image of $\X$ under the associated period mapping 
$\tau$.
\end{notation}

The map $\tau$ is holomorphic. We will refer to the image of $\tau$ in $\Omega$
as the period points. (Occasionally, we will consider $\Omega$ in $L_\C$
instead of $\PP(L_\C)$, and a generator of $\varphi_\C(\h^{2,0}(\X_t))$
as the period vector.)

\begin{defn}
A deformation $p: \X \ra B$, of a K3 surface $X_0:=p^{-1}(b_0)$ is called 
(locally) universal, if (locally) every other deformation $p': \X' \ra B'$ is 
obtained as the pullback from $\X$ by a suitable analytic map $f:B' \ra B$, 
with $f(b_0')=f(b_0)$. 
\end{defn}
We have the following theorem due to Kodaira.
\begin{thm}
Given a K3 surface $X_0$, 
there exists a locally universal deformation $p:\X \ra B$
of the surface $X_0$. 
The associated period mapping $\tau: B\ra \Omega$ is a local isomorphism.
\end{thm}

The family of
deformations of a K3 surface is unobstructed and in particular 20 dimensional.
The K3 surface corresponding to a generic point of $\Omega$ is not algebraic.
The family of algebraic K3 surfaces corresponds to a dense countable union of 
hypersurfaces of the period domain.
See \cite{BW} for more details.



\section{Genus one fibrations}

In this section we explain the Donagi-Pantev result with its applications
to some examples.


\subsection{Examples}

We first give some definitions.

\begin{defn}
A \textbf{genus one fibered} K3 is a K3 surface $X$, which admits a morphism
to $\PP^1$, such that the generic fiber of this morphism is a genus one curve.
\end{defn}

Given a curve $C$ in $X$, the fibre degree of $C$ is defined to be the 
intersection number $C \cdot f$ where $f$ is the generic fibre.
Consider the fibre degree of all curves in $X$.  Then we have the following
definition.
 
\begin{defn}
Given a genus one fibred algebraic K3 surface $X$ the smallest positive
fibre degree is called the {\textbf{index}} of $X$.
\end{defn}

\begin{defn}
A \textbf{multisection} of a fibration is a 
smooth curve in $X$ which intersects 
 each fiber in finitely many points. 
\end{defn}

If $X$ admits a section then the index is equal to 1 and $X$ is called
an {\textbf{elliptic fibration}}.  In the case of algebraic K3s the index
can be viewed as a measure of how far away the surface is from being
an elliptic fibration.



The following are examples of algebraic families
of genus one fibrations.

\begin{defn} \label{defn:M}
Let $\M$ denote the 19 dimensional 
family of K3 surfaces which can be realized as double 
covers of $\PP^2$ branched along a sextic curve $B$. 

\begin{note}
Here $\M$ actually stands for the parameter space of all such K3s.
We will often refer to the K3 surface $M$ corresponding to a generic point
in $\M$ as the generic element of $\M$.

\end{note}

\begin{enumerate} 
\item 
Let $\Msa$ denote the codimension one subfamily of $\M$ consisting of
double covers of $\PP^2$, branched along a sextic curve with at least
one ordinary double point. This contains a Zariski open set $U_{\al}$
of sextics with exactly one node.
\item
Let $\Ma$ be the resolution of this family.
That is, there is
a map\\ $\psi: \Ma \rightarrow \Msa$ where $\psi: \ma \ra \msa$ 
is the resolution of the generic element $\msa$. 


\end{enumerate}
\end{defn} 


Using the adjunction formula, Serre duality and the Riemann-Roch theorem
for surfaces, we get the following results.

\begin{lemma}
The generic element $M_\alpha$ of $\Ma$ is a genus one fibered K3 surface with 
index 2. The Picard group  of $\ma$ is generated by
the pullback of $\Oh_{\PP^2}(1)$  and 
the class of the exceptional curve.
\end{lemma}


\begin{lemma}
Consider the zero locus of a homogeneous equation of bi-degree $(2,3)$ in
$\PP^1\times\PP^2$. It is a K3 surface, which is a double cover of $\PP^2$
branched along a sextic, as well as a genus one fibration of index $3$.
\end{lemma}

Let $\mb$ denote the surface corresponding to the class $(2,3)$ in
$\PP^1\times \PP^2$. 

Projection onto the first factor gives a morphism $\pi_1:\mb \ra \PP^1$,
with the generic fiber a smooth genus one curve corresponding to a cubic
curve in $\PP^2$.

Projection onto the second factor gives a morphism $\pi_2: \mb \ra
\PP^2$, which is generically a $2:1$ map and realises $\mb$
as a double cover of $\PP^2$.

\begin{defn} \label{defn:(2,3)}
Let $\mb$ be a surface in $\PP^1 \times \PP^2$ with divisor class 
$(2,3)$.
Let
$F:=(1,0)_{|{\mb}},\,D:=(0,1)_{|{\mb}}$, where $(1,0)$ and $(0,1)$ denote
 the pullbacks of $\Oh_{\PP^1}(1)$ and  
$\Oh_{\PP^2}(1)$ respectively.
\end{defn}

Then $\Pic(\mb)=\Z\,D\oplus\Z\,F$ where
$\intf{D,D}=2, \intf{F,F}=0, \intf{D,F}=3$.

The generic curve in $|D|$ is a multisection of fiber degree 3.
Therefore $\mb$ is a genus one fibration of index 3.



Let $\Mb$ denote the family of all K3 surfaces obtained in this way.
The generic element $M_\be$ embeds in $\PP^5$ as a surface of 
degree 8.


Similarly we Let $\Y$ denote the 19 dimensional family of 
degree 8 K3 surfaces in 
$\PP^5$. The generic element
$Y$ of $\Y$ is a complete intersection of three quadric hypersurfaces.

The family $\M_\be$ is a codimension one subfamily of $\Y$, and $\M_\al$
is a codimension one subfamily of $\M$.


%The inclusion relations between the families can be summarized as follows:
%\begin{displaymath}
%\begin{array}[5]{lllll}
%\\Ma & \subset & \M \\
%\\Mb & \subset & \Y \\
%\end{array}
%\end{displaymath}


\subsection{Ogg-Shafarevich Theory}
Given a smooth genus one fibration $X\ra  B$, there is a naturally
associated elliptic fibration $J \ra B$, called the relative Jacobian
of $X$. More precisely we have the following definition.

\begin{defn}
Let $p_X: X\ra B$ be a genus one fibration. Fix a relatively ample sheaf
$\Oh_X(1)$. Let $p_J:J\ra B$ be the relative moduli space of semistable
torsion-free sheaves of rank 1, degree 0 on the fibers of $X$. The
fibration $p_J:J\ra B$ is called the relative Jacobian of $X\ra B$. 
\end{defn}

\begin{note}
We will sometimes denote $p_J:J\ra B$, by $J(X)$.
\end{note}
For existence, see \cite{FM}.

An elliptic fibration $p_J:J \ra B$ can be the relative Jacobian 
of different genus one fibrations. One way to distinguish them is 
by the index. In the case of algebraic K3 surfaces, the index is always
finite. For example the generic element $\ma \in \Ma$ is a genus
one fibred K3 of index 2, while $\mb \in \Mb$ has index 3.

\begin{defn}
There are two natural invariants associated to a genus one fibration
$p_X: X\ra B$. Let $f_b$ denote the fiber over $b\in B$.
\begin{enumerate}
\item {\textbf {$j$-function:}} The   $j$-function of $X$ denoted by $j_X$ is
a function $j_X: B\ra \PP^1$ defined by
$$ j_X: b \ra j(f_b)$$,
where $j(f_b)$ is the $j$-invariant of $f_b$.
\item \textbf{Monodromy Presentation}:
Let $\{b_1, \ldots, b_k\}$ (also known as the discriminant) denote the points 
over
which the fibers are singular. The
 monodromy presentation of $X$ is a homomorphism 
$$\rho_X: \pi_1(B-\{b_1, \ldots, b_k\}) \ra SL(2,\Z).$$
\end{enumerate}
\end{defn}

Let $J\ra \PP^1$ be an elliptic surface (i.e. it has a section). 
Then associated
to $J$ there is another elliptic surface $\overline{J}\ra \PP^1$, called the 
\textbf{Weierstrass model} of $J$. It is uniquely determined 
up to fibre preserving automorphisms of $J$.

We have the following theorem by Friedman  
on the moduli space of elliptic K3 surfaces
(See \cite{FM}, Proposition 4.3).

\begin{thm} \label{thm:fromFM}
Let $J\ra \PP^1$ be an elliptic K3 surface. Then $\overline{J}$ 
is determined by the
choice of two sections $g_2\in \h^0(\PP^1,\Oh(8))$ and 
$g_3\in \h^0(\PP^1,\Oh(12))$, modulo the action of $\C^*$.
 
We identify two elliptic surfaces over $\PP^1$ if there is an isomorphism
between them covering some automorphism of the base, $\PP^1$.
The moduli
space of such surfaces(denoted by $\J$) has dimension 18. 
Let $U\subset \J$ be the set corresponding to the pairs $(g_2,g_3)$ such that
\begin{enumerate}
\item $g^3_2-27g^2_3$ is a nonzero section of $\Oh(24)$,
\item
For all $p \in \PP^1$, $\min\{3v_p(g_2),2v_p(g_3)\}<12$, where $v_p$ denotes 
the order of vanishing of the corresponding section at $p$.
\end{enumerate}



Then $U$ is Zariski open, and non-empty. Conditions 1) and 2) ensure that
the generic fiber of $p_J: J\ra \PP^1$ is smooth and that $J$ has at most
ordinary double points.
\end{thm}
The generic point $[J]$ in the moduli space corresponds to a smooth surface
$J$ with 24 nodal fibers. Then $j_J$ has degree 24 ($j_J$ maps the discriminant
to the point $\infty$) and the monodromy presentation is a homomorphism
$$\rho_J: \pi_1(\PP^1-\{b_1, \ldots, b_{24}\}) \ra SL(2,\Z).$$

If $X$ is a genus one fibered K3 surface, then $X$ has the same $j$-function
as its relative Jacobian $J(X)$, and its monodromy presentation $\rho_X$ 
is isomorphic to $\rho_J$, via conjugation.

The classical theory of elliptic fibrations has been developed by Ogg
and Shafarevich. For details of the technical aspects, see \cite{DG}.


The main idea is that given an elliptic fibration $p_J:J\ra B$, the set
of all genus one fibrations $p_X:X\ra B$ whose relative Jacobian is
isomorphic to $J$ is in one-to-one correspondence with the
Tate-Shafarevich group $\sha(J)$ of $J$. 

\begin{defn}
Given an elliptic fibration $p_J:J\ra B$, the Tate-Shafarevich group
$\sha(J)$ is defined to be $\h^1(B,J^\#)$ where $J^\#$ is the sheaf of
local sections of $p_J:J\ra B$. 
\end{defn}

As mentioned earlier, all elements of $\sha(J)$ have the same $j$-function
and isomorphic monodromy presentation.

This makes sense in any topology. Of interest to us are the \'etale and
analytic topologies. However, the algebraic genus one fibrations
associated to $J$ correspond only to $\h^1_{\text{\'et}}(B,J^\#)$. (See
\cite{DP} ).

The Ogg-Shafaravich Theory of elliptic fibrations is defined for any
dimension. In the case that $J$ is an elliptic K3 surface, we get a simple
characterization of $\sha(J)$.

\begin{lemma}
Let $J\ra\PP^1$ be an elliptic K3 surface. Then
$$\sha(J)=\Br(J)=T^\vee_J\tensor\Q/\Z.$$
\end{lemma}
\begin{proof} See \cite{DG}. \end{proof}

C\u ald\u araru shows in his thesis that $J$, viewed as a relative moduli
space of sheaves on $X$, is not a fine moduli space. In fact, if the
surface $X$ corresponds to the element $\alpha\in \Br(J)$, then there
exists a $p^*_2(\alpha)$-twisted Poincare sheaf $\E$ on $X\times J$,
where $p_1,p_2$ are the first and the second projections.

This twisted Poincare sheaf $\E$ induces a map on the level of 
cohomology from $\h^*(X,\Q)
\ra \h^*(J,\Q)$ called the Fourier-Mukai Transform which is a Hodge
isometry of the extended Mukai product (see Section 5).

As a result, we get a short exact sequence  
$$0 \ra T_X \ra T_J \stackrel{\alpha}{\ra} \Z/n\Z \ra 0.$$
In other words, $T_X$ embeds in $T_J$ as the kernel of $\alpha$.

Let $\xa,\yb$ be two genus one fibrations associated to the elliptic
fibration $J$. Then the index of $\txa$ (resp. $\tyb$) is equal to 
$[T_J:\txa]$ (resp. $[T_J:\tyb]$).





Let $\ob:=\beta_{|\txa}$ and $\oa:=\alpha_{|\tyb}$. Then
$\ker\ob=\ker\alpha\cap\ker\beta=\ker\oa$. So $\ker\ob$ is Hodge
isometric to $\ker\oa$. Donagi and Pantev have proved the following
 interesting
case of C\u ald\u araru's conjecture.

\begin{thm} \label{thm:D-P}
Let $J/S$ be an elliptic K3 surface with a section, and let $\alpha, \beta
\in Br(J)$.  Identifying $Br(J)$ with the Tate-Shafarevich group of $J$
yields genus one fibered K3 surfaces $\xa, \yb$ 
(in general without a
section) which correspond to $\al, \be$ respectively.  Viewing $J$ as
a moduli space of stable sheaves on $\xa$ and $\yb$ gives
surjections $Br(J) \ra Br(\xa)$ and $Br(J) \ra Br(\yb)$; let
$\bar{\be}$ be the image of $\be$ in $Br(\xa)$ and
$\bar{\al}$ be the image of $\al$ in $Br(\yb)$.  Then there
exists an equivalence of derived categories of twisted sheaves $\mathcal
{D_{coh}}^{b}(\xa, \bar{\be})$ and $ \mathcal
{D_{coh}}^{b}(\yb, \bar{\al})$.
\end{thm}

We will now apply the Donagi-Pantev result to the families
$\Ma$ and $\Mb$ discussed earlier.
They are 18 dimensional algebraic families of 
genus one fibered K3 surfaces, whose generic elements $\ma, \mb$ are not
elliptically fibred. Instead they have multisections of index 2 and 3
respectively.

\begin{defn}
Let $\Ua$ (resp. $\Ub$) in $\Ma$ (resp. $\Mb$) denote the Zariski
open subset corresponding to smooth genus one fibrations with 24 nodal fibers.
\end{defn}

Then we have the following theorem:


\begin{thm} \label{thm:Ma-Mb}
There exists a finite to finite correspondence between the families $\Ma$ and
$\Mb$, over the open sets $\Ua\subset \Ma, \Ub\subset \Mb$. 
That is for each element $\ma\in\Ma$, there exist finitely many elements 
$\{ M^j_\beta\}^l_{j=1} \subset \Mb$, with isomorphic relative Jacobian.
\end{thm} 

\begin{proof}
The main idea is to show the classifying morphisms from $\Ua$
and $\Ub$ to $\J$ are dominant maps between algebraic varieties of
the same dimension.  Then it follows that they have finite fibres.


As in Theorem \ref{thm:fromFM}, let $U\subset \J$ denote 
the Zariski open irreducible
subset corresponding to smooth elliptic fibrations with 24 nodal fibers.
Let $p_J: J\ra \PP^1$, be a representative of $[J] \in U$.
Let $j_J$ denote the
$j$-function of $J$, and let $\rho$ denote the monodromy 
representation of $J$.

Let $\pha: \Ma \dashrightarrow \J$ denote the rational map sending a
fiber $\ma$ to the isomorphism class of its relative Jacobian, $[J(\ma)]$.
Similarly, let $\phb: \Mb \dashrightarrow \J$ denote the classifying
morphism for the family $\Mb$.

Since $\Ua$ is an irreducible Zariski open subset of $\Ma$, the image of $\Ua$
under $\pha$ is an irreducible subvariety of $U \subset \J$.

Fix $[J]\in \pha(\Ua)$. 

Then all the elements in 
$\varphi^{-1}_\alpha[J]$ have the same $j$-function and isomorphic monodromy
representation. The set 
$\Hom ( \pi( \PP^1-\{d_1, \ldots, d_{24}\}),SL(2,\Z))$ is
discrete 
hence so is the orbit of $\rho_J$ under conjugation, which is the set
of all  
 representations isomorphic to $\rho_J$.
 
 It follows that the preimage $\varphi^{-1}_\alpha[J]$ in $\Ua$ is discrete.
 So the image of $\Ua$ in $\J$ is 18 dimensional, and hence an irreducible
 Zariski open subset (i.e. $\pha$ is dominant). 
The same argument applies to $\phb$. 
 
Since $\pha_{|\Ua},\phb_{|\Ub}$ are algebraic maps between irreducible
varieties of the same dimension, they have finite fibers.
So given any point $[J]$ in the image of $\pha$ and $\phb$
there exist finitely
many elements $\{M^i_\alpha\}^k_{i=1}, \{M^j_\beta\}^l_{j=1} \in \Ma, \Mb$
respectively with their relative Jacobian isomorphic to $J$.
\end{proof}

This then gives us 18 dimensional loci in $\Ma \times \Mb$ of pairs
$(\ma,\mb)$ with isomorphic jacobians.
Recall that $\tma$ and $\tmb$
embed in
$T_J$ as $\ker\alpha$, $\ker\beta$ respectively for some 
$\alpha,\beta \in \Br(J)$ and that
$\ker\beta_{|\tma}\cong \ker\alpha_{|\tmb}$. 
The Donagi-Pantev result then 
implies that $$\D(\ma,\ob) \cong \D(\mb,\oa).$$
So we have an explicit example where C\u ald\u araru's Conjecture
holds.

There is a natural corollary to this theorem.

\begin{corollary} \label{cor:M_i-M_j} 
The correspondence of Theorem~\ref{thm:Ma-Mb} above implies 
that for each $i$ and $j$, there exists an
algebraic cycle $Z^{ij}\in \h^{2,2}(M^i_\alpha\times M^j_\beta,\Q) $ such
that $$Z^{ij}:\h^2(M^i_\alpha,\Q) \ra \h^2(M^j_\beta,\Q)$$ is a Hodge
isometry.
\end{corollary}
\begin{proof}
Since $J$ is the relative moduli space of sheaves on 
$M^i_\alpha$ and $M^j_\beta$ for each $i,j$ $(1\leq i \leq k,\, 1\leq j \leq
l)$, there exist maps
$$0 \ra T_{M^i_\alpha} \ra T_J \ra \Z/2\Z \ra 0$$,
$$0 \ra T_{M^j_\beta} \ra T_J \ra \Z/3\Z \ra 0.$$
This implies that there exists a Hodge isometry 
$\psi^{ij}: (T_{M^i_\alpha})_\Q \ra (T_{M^j_\beta} )_\Q$. 
Let $Z^{ij}$ denote the
corresponding element of $(T_{M^i_\alpha} \tensor T_{M^j_\beta} )_\Q$.

%Recall that $\Pic(M^i_\alpha)=\Z\,D\oplus\Z\,f$ and
%$\Pic(M^i_\alpha)=\Z\,D\oplus\Z\,F$.

%Consider the element  $Z^{ij}_N:=\frac{2}{3} D\tensor F+\recp{4}f\tensor
%D-\frac{5}{12} f\tensor F$ in $(N_{M^i_\alpha} \tensor N_{M^j_\beta} )_\Q$. 
%Then $Z^{ij}:=Z^{ij}_N+Z^{ij}_T$ induces a Hodge isometry 
%$Z^{ij}: \h^2({M^i_\alpha},\Q) \ra \h^2({M^j_\beta}, \Q)$.

By the following recent theorem of Mukai (see \cite{M3}), it follows that
$Z^{ij}$ is algebraic.
\begin{thm} \label{thm:Mu}
Let $X$ and $Y$ be algebraic K3 surfaces. Suppose we are given a map
 $f:\h^2(X,\Q) \ra \h^2(Y,\Q)$
which is a Hodge isometry. Then the associated cycle $Z_f$ is algebraic.
\end{thm}

\end{proof}


Our goal is to find other loci consisting of
K3 surfaces exhibiting a similar
correspondence.
These will provide a suitable framework for extending the
Donagi-Pantev
result to non fibred K3 surfaces. 

In the next section we will exhibit such correspondences 
between non-fibred K3
surfaces.

We remark here that the Donagi-Pantev theorem can be viewed as
 a statement about T-duality on
gerbes over K3s.

\section{Families of correspondences}

In this section we show the existence of infinitely many families
of pairs of corresponding K3 surfaces.  We first start with some 
examples.

\subsection{Examples}

Theorem \ref{thm:Ma-Mb} implies that there exists an 18 dimensional locus 
$\F \subset \Ma \times \Mb$ with the following properties:


\begin{enumerate}
\item
Given $(\ma, \mb)$ corresponding to points in $\F$, there exists
an algebraic class $Z$ in $\h^{2,2}(M_\alpha\times M_\beta, \Q)$
such that the associated map \\
$Z:\h^2(M_\alpha,\Q) \ra \h^2(M_\beta,\Q)$
is a Hodge isometry,
\item
The pairs $M_\alpha$, $M_\beta$ correspond to elements $ \alpha,\beta
\in \Br(J)$ resp. where $J$ is the associated relative Jacobian fibration.
Recall that $\tma$ embeds in $T_J$ as $\ker \al$ and
 $\tmb$ embeds in $T_J$ as $\ker \be$.
\end{enumerate}

Let $\overline{\alpha}:=\beta|_{T_{M_\alpha}}, 
\overline{\beta}:=\alpha|_{T_{M_\beta}}$, be the corresponding elements
in $\Br(\ma)$ and $\Br(\mb)$.

Then $\ker\overline{\alpha}$ is Hodge isometric to 
$\ker\overline{\beta}$, via $Z$.

%Witt's thm
%For future use we quote here Witt's Theorem on symmetric forms 
%without proof. For a proof,
%see \cite{L}.
%\begin{thm}
%Let $E$ be a vector space over a field $k$ ($\textnormal{char} k \neq 2$) 
%and let $g$ be a non-degenerate symmetric form on $E$. Let $F,F'$ be subspaces
%of $E$, and let $\sigma: F\ra F'$ be an isometry. Then $\sigma$ can be 
%extended to an isometry of $E$.
%\end{thm}

We would like to extend the above isometry  to other elements of $\M\times\Y$.

\begin{thm} \label{thm:M-Y}
There exists a 19 dimensional family in $\M\times\Y$ of isogenous pairs
$(M,Y)$ such
that $\tmq$ is Hodge isometric to $\tyq$.
\end{thm}


\begin{proof}[Proof of Theorem \ref{thm:M-Y}]
Consider a marked deformation of $\ma$ inside the family $\M$.
%i.e. choose an open set $U$ in $\Omega_{\M}$
%such that there exists an isomorphism $\varphi: R^2 \tau_*\Z_{\M} 
%\ra L \times U$. 
If we choose $U \subset \M$ small enough so that it is contractible then such 
a marking exists. For $M$ a general element of $U$
we get an inclusion\\
$i: \tma \ra \tm$.

%The isomorphism $\varphi$, gives us an identification of $\h^2(M,\Z)$
%with $\h^2(X'_\alpha,\Z)$ for each fibre $X$. 

Recall that $N_\ma =\Z\,D \oplus \Z\,e$ where 
$$\intf{D,D}=2,
\intf{D,e}=0, \intf{e,e}=-2.$$
Since $N_M:=\Z\,D$, $N_M^\perp$ in $N_\ma$ is just $\Z\,e$.
Hence we get an inclusion $i: \tma \oplus \Z\,e \ra \tm$
 as a sublattice of 
index 2. 



Next, consider a marked deformation of $\Mb$ in the family
$\Y$. Let $Y$ be the generic element.  Let $H$ denote the hyperplane 
class of $Y$.  When restricted to a generic 
element $\mb$ in the subfamily $\Mb$, $H = D + F$.
As before $N_{\mb} =\Z\,D \oplus \Z\,F$, where $$\intf{D,D} = 2,
\intf{D,F} = 3, 
\intf{F,F} = 0.$$  Since $N_Y = \Z\,H$ and $H = D+F$ on a generic
element $\mb$ we have $N_Y^{\perp} =\Z\,(3H-8F)$ in $N_{\mb}$.
Let $r:= 3H-8F$, then we have an inclusion:
$i: \tmb \oplus \Z\,r \ra \ty$ as a sublattice of index 9.

%Let $L_\alpha, L_\beta$ denote the images of 
%$\ker\intf{\frac{\overline v_\beta}{3},\cdot},
%\ker\intf{\frac{\overline v_\alpha}{2},\cdot}$ in $T_M, T_{Y}$
%respectively.

According to Corollary \ref{cor:M_i-M_j} we
 already have a Hodge isometry between $(\tma)_\Q$ and 
$(\tmb)_\Q$ induced by a correspondence $Z_T'$.
We would like to know when we can extend this isometry to other elements
of the families $\M, \Y$.



Let $V:=\tmq, W:=\tyq$.\\
Let $V':=(\tma)_{\Q}, W':=(\tmb)_{\Q}$.\\
It follows that $V = V'\oplus \Q\,e$,
$W = W'\oplus \Q\, r$
and $Z'_T: V' \ra W'$ is an isometry.

Consider the following element of 
${V}^{*} \tensor W$,\\
$Z_T:=Z'_T-\frac{1}{12} e \tensor r$. Then it is easy to check that
$Z_T$ induces an isometry between $V$ and $W$.


Since $\tm$ and $\ty$ are transcendental lattices of K3s,
 $V$ and $W$ both embed in $L_\Q$. 


Now let $\omega_M \in V_\C$ be the associated
 period point of $M$.  Since $Z_T$ is a $\Q$ isometry, 
the image $Z_T(\omega_M)$ under its 
$\C$-linear extension lies in the period domain $\Omega$.

\begin{note}
We have the following relations 
$$\intf{Z_T(\omega_M),Z_T(\omega_M)}=0$$ and
$$\intf{\omega_M, \overline\omega_M}=
\intf{Z_T(\omega_M),Z_T(\overline\omega_M)}>0$$
which are the relations satisfied by points in $\Omega$.
\end{note}

By the surjectivity of the period mapping we get that
$Z_T(\omega_M)$ is the period point $\omega_Y$ of some K3 surface $Y$.
For this choice of $Y$, 
$Z_T: \txq \ra \tyq$ becomes a Hodge isometry. 

It is easy to extend this isometry to all of $\h^2(M)_\Q$.
Consider the cycle
$Z:=\frac{1}{4} D \tensor H + Z_T  \in \h^2(M,\Q) \times \h^2(Y,\Q)$.
 Then $Z$ maps the polarisation on $M$ to a rational multiple of
 the polarisation on $Y$
and restricts to the Hodge isometry $Z_T: \tmq \longrightarrow  \tyq$. 
Hence it induces a Hodge isometry $$Z: \h^2(M,\Q) \longrightarrow
\h^2(Y,\Q).$$
By Mukai's result it now follows that $Z$ is algebraic.
This proves Theorem~\ref{thm:M-Y}.

\end{proof}


\begin{note}
The Hodge isometry $Z_T:\tmq \longrightarrow \tyq$ is strictly 
defined over $\Q$ because its component $Z_T'$ is strictly defined over
$\Q$.  
\end{note}



\subsection{General construction}

We will now generalise the above example.

\begin{lemma}
Let $\Xkkk, \Xk, \Xkk$ denote the 19 dimensional families of algebraic K3s 
in $\PP^{3k}, \PP^{3k+1}, \PP^{3k+2}$ respectively. \\
 Let $\xkkk \in \X_{3k}, X_{3k+1} \in \X_{3k+1}, X_{3k+2} \in \X_{3k+2}$ 
denote the general elements
with polarisations $D_{3k}, D_{3k+1}, D_{3k+2}$ respectively.  Then
$$\intf{D_{3k},D_{3k}} = 6k-2,\intf{D_{3k+1},D_{3k+1}} = 6k,
\intf{D_{3k+2}, D_{3k+2}} = 6k+2.$$  
We consider the following 18 dimensional families of genus one fibred K3s:
\begin{enumerate}
\item
Let $\J_{0}$ be the family of quartic surfaces in $\PP^3$ containing a line.
\item
Let $\J_{1}$ be the family of smooth K3s given by the intersection of a 
smooth cubic hypersurface with a nodal quadric.
\item
Let $\J_{2}$ be the family of genus one fibred K3s corresponding to the 
class of $(2,3)$ in $\PP^1 \times \PP^2$.



\end{enumerate}
Then we have the following inclusions
$$\J_{0} \subset \Xkkk, \J_{1} \subset \Xk, \J_{2} \subset \Xkk.$$

\end{lemma}

The proof follows from an application of the Riemann-Roch theorem 
to show that linear systems of the form $H + (k-1)E, k\geq 1$, where $H$
represents the hyperplane class and $E$
represents an elliptic curve, are very ample.  


\begin{thm} \label{thm:fa-Xk}
There are infinitely many pairs of integers $(k,l)$ and $(m,l)$,
satisfying certain constraints 
for which there exist 19 dimensional loci of correspondences 
in $\X_{3k} \times \X_{3l+1}$ and 
$\X_{3m+2} \times \X_{3l+1}$. That is, there are 19 dimensional 
families of pairs
$(\xkkk, \xl)$ and $(\xmm, \xl)$ on which there
exist algebraic correspondences inducing Hodge isometries:
$Z_{3k,3l+1} : \h^2(\xkkk, \Q) \rightarrow \h^2(\xl, \Q) $ 
and $Z_{3l+1, 3m+2} : \h^2(\xmm, \Q) \rightarrow \h^2(\xl, \Q)$.
 \end{thm}

\begin{proof}
The idea is the same as in Theorem~\ref{thm:M-Y}.  
There exists a finite to finite correspondence
between $\J_{0}$ and $\J_{1}$ (Theorem~\ref{thm:Ma-Mb}).  
Let $J_{0} \in \J_{0}$
and $J_{1} \in \J_{1}$ be genus one fibred K3s with isomorphic Jacobians.
By Corollary~\ref{cor:M_i-M_j} there exists an algebraic cycle
$Z'_T \in (T_{J_{0}} \tensor T_{J_{1}})_\Q$ such that 
$Z'_T: (T_{J_{0}})_{\Q} \rightarrow (T_{J_{1}})_{\Q}$ is a Hodge isometry.

Next consider a marked deformation of $J_{0} \in \Xkkk$.  Then 
$T_{J_{0}} \rightarrow T_{\xkkk}$ where $\xkkk$ is a generic fibre.
Let $D_{3k}$ denote the hyperplane class of $\xkkk$.  Similarly
consider a marked deformation of $J_{1} \in \X_{3l+1}$.  Again for 
$\xl$ a generic element, $T_{J_{1}} \rightarrow T_{\xl}$ and the 
polarisation $D_{3l+1}$ is the orthogonal complement of $T_{\xl}$ 

Consider the following extension $Z'$ of $Z'_T$:
$$Z':= Z'_T + \frac{1}{\lambda}D_{3k} \tensor D_{3l+1}.$$  Then 
$Z'$ restricts to $Z'_T$ on
 $(T_{J_{0}})_{\Q} \rightarrow (T_{J_{1}})_{\Q}$ and maps
$D_{3k}$ to $ \frac{1}{\lambda} \intf{D_{3k},D_{3k}}D_{3l+1}$
for some $\lambda$.
If $D_{3k}$ maps isometrically to a rational multiple of $D_{3l+1}$
i.e. if ${\lambda}^2 = \intf{D_{3k}, D_{3k}} . \intf{D_{3l+1, 3l+1}}$,
then by Witt's theorem (see \cite{L}), $Z'$ extends to an isometry 
$Z:\h^2(\xkkk, \Q) \ra \h^2(\xl, \Q)$.  In this case  
$Z :(D_{3k})_{\Q} \rightarrow (D_{3l+1})_{\Q} $, hence 
$(D_{3k}^\perp)_{\Q}$ maps to $(D_{3l+1}^\perp)_{\Q}$, that is
we get an isometry of $\Q$ vector spaces:
$Z_T:(T_{\xkkk})_{\Q} \rightarrow (T_{\xl})_{\Q}$.  As before
consider the period point $\omega_{\xkkk} \in (T_{\xkkk})_{\C}$
of $\xkkk$.  Since $Z_T$ is an isometry its image 
$Z_T(\omega_{\xkkk})$ will lie in the period space.  By surjectivity
of the period mapping it is the period point of 
some other K3 surface in $\Xl$.  We choose that K3 surface $\xl$
to be the corresponding partner of $X_{3k}$.  Then for this choice
of $X_{3l+1}$, $Z_T:(T_{\xkkk})_\Q \ra (T_{X_{3l+1}})_\Q.$  
In this way we get a
family of corresponding pairs $(\xkkk, \xl)$.  

However in order for this to work  we need to solve for $\lambda$
such that
$ {\lambda}^2 = \intf{D_{3k},D_{3k}} . \intf{D_{3l+1}, D_{3l+1}}.$
This imposes some constraints on $k$ and $l$.
In other words we need to solve ${\lambda}^2 = (6k-2)(6l)$
which reduces to solving $n^2 = (3k-1)(3l)$, since
we can then just take $\lambda = 2n$.
Given any k, the integer $3k-1$ has a unique prime factorisation.
Notice that it will not contain 3 as a factor since 
$3k-1 \cong 2 (\mod 3)$.  We separate the even powers and let $\rho$
denote the product of the remaining factors.  Then choosing
$l = 3 \rho d^2$ (where $d \in \Z$) makes 
$(3k-1)(3l)$ a perfect square.  Since for any $k$ there are infinitely
many such $l$ we get infinitely many families of correspondences.

In a similar way one shows the existence of loci of correspondences
in $\Xmm \times \Xl$.  It amounts to solving for $m$ and $n$
such that $(3m+2)(3l) = n^2$.  The previous argument can be
applied to show that such $m$ and $n$ exist.

\end{proof}

Note that we don't get correspondences between generic elements
of $\Xkkk, \Xmm$.  For those to exist we need
 $(3k-1)(3m+2) = n^2$.  The RHS can only be congruent to 0 or 1 $(\mod 3)$
whereas the left hand side is equal to 2 $(\mod 3)$, so there are no 
solutions.


The genus one fibrations of the above example all had index 3.
We can get correspoondences based on fibrations of index 2
as well.  Let $\J_3$ be the 18 dimensional family of K3s
realised as double covers of $\PP^1 \times \PP^1$ branched 
along a $(4,4)$ curve.  Then a generic element $J_3 \in \J_3$
has index 2 and embeds in $\PP^{4m+5}$ for $m \geq 1$. 
Let $\Y_{4m+5}$ denote the family of algebraic K3s in $\PP^{4m+5}$.
By the same arguments as in the proof of Theorem~\ref{thm:fa-Xk}
one can show the following.






\begin{thm}
There are infintely many pairs of integers $(k,m)$
for which there
are 19 dimensional families of correspondences in 
$$\Xkkk \times \Y_{4m+5},\quad\Xk \times \Y_{4m+5},\quad and\quad
\Xkk \times \Y_{4m+5}.$$
\end{thm} 

\begin{note}
The $(k,m)$ for each family will be different.
\end{note}

The question of whether these $\Q$ isometries correspond to appropriate
elements $\al$ and $\be$ in the Brauer groups of the 
corresponding surfaces is currently under investigation.



\section{A classical correspondence}


In this chapter we consider the moduli space of sheaves on a K3 surface.
We present the results of Mukai and C\u ald\u araru, and prove a theorem 
pertaining to a classical correspondence.

\subsection{Moduli spaces}


\begin{defn}
Let $Y$ be a surface, $\Oh_Y(1)$ be an ample class on $Y$, $M$ the moduli
space of $\Oh_Y(1)$-semistable sheaves with prescribed Chern classes. Then
$M$ is a \textbf{fine moduli space} if there exists  a universal sheaf $\E$
on $Y\times M$ such that $\E_{|Y\times m}\cong E_m$, where $E_m$ represents
an element of the isomorphism class of sheaves corresponding to the point 
$m\in M$. 
\end{defn}

In general $M$ is not a fine moduli space, i.e. a universal sheaf does
not exist on $Y\times M$. 
However, another sheaf closely related to the universal sheaf does exist.

\begin{defn} \label{defn:QuasiUniv}
Let $Y$ and $M$ be as in the definition above. Then a quasi-universal 
sheaf is a sheaf $\E$ on $Y\times M$, flat over $M$ such that if
$m\in M$ represents the isomorphism class of the sheaf $E_m$ on $Y$,
then $\E_{|Y\times m}\cong E_m^n$ for some $n>0$. 
\end{defn}



Mukai proved that a quasi-universal sheaf associated to a moduli problem
always exists. (See \cite{M1}, Appendix 6).

In the next section, we will explicitly construct the quasi-universal 
sheaf for a classical moduli problem.



Consider $\E$ on $Y\times M$ and let $p_Y, p_M$ denote the first and the
second projections. Set 
$$Z_\E:=\frac{p_Y^*(\sqrt{\td_Y})\cdot \ch(\E)\cdot p_M^*(\sqrt{\td_M})}
{n},$$
where $n$ is as in Definition~\ref{defn:QuasiUniv}.

$Z_\E$ is an algebraic cycle on $Y\times M$ and induces a homomorphism
$$
\begin{array}{ccccc}
Z_\E &: &\h^*(X,\Q) & \longrightarrow & \h^*(M,\Q) \\
&& t & \mapsto & p_{M*}(p_Y^*(t)\cdot Z_\E)
\end{array}
$$

\begin{defn}
The \textbf{Mukai vector} $v$ associated to a sheaf $F$ on a K3 surface 
$Y$ is given by
$v(F):=\ch(F)\cdot\sqrt{\td_Y}$. It decomposes as a sum $v=a^0+a^2+a^4$, where 
$a^i\in\h^{2i}$, 
also denoted by $(a^0,a^2,a^4)$ at times.
\end{defn}

\begin{defn}
The \textbf{extended Mukai lattice} is defined to be 
$\widetilde{\h}(Y,\Z):=\h^0(Y,\Z)\oplus\h^2(Y,\Z)\oplus\h^4(Y,\Z)$
with the pairing 
$$\intf{(a^0,a^2,a^4),(b^0,b^2,b^4)}=
\intf{a^2,b^2}-\intf{a^0,b^4}-\intf{a^4,b^0}$$
where $\intf{,}$ stands for the cup product. The extended Mukai lattice
has a natural weight 2 Hodge structure
$$
\begin{array}{lcl}
\widetilde\h^{2,0}(Y,\C) & := & \h^{2,0}(Y,\C) \\
\widetilde\h^{1,1}(Y,\C) & := & 
\h^0(Y,\C)\oplus\h^{1,1}(Y,\C)\oplus\h^4(Y,\C) \\
\widetilde\h^{0,2}(Y,\C) & := & \h^{0,2}(Y,\C) 
\end{array}
$$

\end{defn}

\begin{defn}
An element $v$ in $\tilde\h(Y,\Z)$ is a \textbf{primitive Mukai vector} if
$v$ is not a multiple of any other element of $\widetilde\h(Y,\Z)$. The vector
$v$ is called isotropic if $\intf{v,v}$=0.
\end{defn}

\begin{remark}
Another characterization of $n$ is as follows: Let $v$ be a primitive
Mukai vector in $\widetilde\h(Y,\Z)$. Then $n=\gcd\{ \intf{\omega,v} \}$,
where $\omega$ runs over all elements of $\widetilde\h^{1,1}(Y,\Z)$.
For a proof, see \cite{M1}, Appendix 6.
\end{remark}

\begin{defn}
Let $Y$ be an $n$-dimensional algebraic variety. Let $\Oh_Y(1)$ be an 
ample class. For a coherent sheaf $F$, let $r(F)$ denote the rank of $F$.
A torsion-free coherent sheaf $E$ on $Y$ is \textbf{(semi)stable} with respect
to $\Oh_Y(1)$ in the sense of Gieseker if  for 
every proper non-zero subsheaf
$F$ of $E$, the following inequality holds 
$$h^0(E\tensor \Oh_Y(k) )/ r(E) >  h^0(F\tensor \Oh_Y(k) )/ r(F)$$
for $k \gg 0$. 
\end{defn}

Now we can formulate a result due to Mukai (see \cite{M2}).

\begin{thm}
Let $Y$ be an algebraic K3 surface and $v$ a primitive isotropic Mukai
vector. Let $M$ be the moduli space of $\Oh_Y(1)$-semistable sheaves with
Mukai vector $v$. Assume that $M$ is nonempty and compact. Then $M$ is
a K3 surface and the cycle $Z_\E$ induces a Hodge isometry $Z_\E:
v^\perp/\Z\,v \ra \h^2(M,\Z)$.
\end{thm}

The transcendental lattice $\ty$ is contained in $v^\perp$ and $\ty\cap
\Z\,v=0$, therefore $v^\perp/\Z\,v$ contains a lattice isomorphic to $\ty$.
From the arguments in the proof of Proposition~{6.4} in \cite{M2},
it follows that $Z_\E$ embeds $T_Y$ in $T_M$ as the kernel of some nontrivial 
element
$\alpha \in T^\vee_M\tensor \Q/\Z$. In other words, we
 have the following exact sequence
$$0\ra T_Y \stackrel{Z_\E}{\ra} T_M \stackrel{\alpha}{\ra} \Z/n\Z \ra 0,$$
where $n$ is as in Definition~\ref{defn:QuasiUniv}.

Let $Y$ be a generic element of the family $\Y$
of degree 8 surfaces in $\PP^5$. Then $Y$ is a transverse intersection 
of three smooth quadric hypersurfaces $Q_0,Q_1,Q_2$. We consider the moduli
space of $\Oh_Y(1)$-semistable sheaves on $Y$ with the following invariants
$c_0=2, c_1=\Oh_Y(1), c_2=4$. The Mukai vector of a sheaf on $Y$ with 
the above invariants is $v=(2,\Oh(1),2)$. Since $\Oh_Y(1)$ is primitive, $v$
is also primitive and
$\intf{v,v}=\Oh(1)\cdot\Oh(1)-2\cdot 2 -2\cdot 2=8-8=0$. It follows by Mukai's 
theorem that if $M$ is non-empty and compact, then $M$ is a K3 surface. 

We will see later that $M$ is isomorphic to a double cover of $\PP^2$ branched
along a sextic.

We prove the following theorem:

\begin{thm}
Let $Y$ be a generic intersection of three smooth quadric hypersurfaces
 in $\PP^5$, $M$ the moduli 
space of $\Oh_Y(1)$-semistable sheaves on $Y$ with Chern classes
$c_0=2, c_1=\Oh_Y(1), c_2=4$, then there exists a rank 4 bundle $\E$ on 
$Y\times M$ such that $\E_{|Y\times m}\cong E_m\oplus E_m$, where 
$E_m$ represents an element of the isomorphism class of sheaves corresponding
to $m\in M$.
\end{thm}

In certain special cases when the rank of $\Pic(Y)\geq 2$, for example when $Y$
contains a line, $M$ is a fine moduli space, i.e. 
a universal sheaf exists on $Y \times M$. 
Before we formulate the actual construction, we need some preliminaries.

\subsection{Preliminaries}
\label{S:pre}

Let Y be the triple intersection of three smooth quadric hypersurfaces 
$Q_0$,$Q_1$ and $Q_2$ in $\PP^5$ with the assumption  
that every quadric containing $Y$ is
of rank $\geq 5$. This is generically the case.

Let $N:=\{ Q: Q:=a_0Q_0+a_1Q_1+a_2Q_2=0 \,\textnormal{in}\, \PP^5 \}$ be the
net of quadrics in $\PP^5$ spanned by $ Q_0,Q_1,Q_2$. Then the set of 
singular members of $N$ corresponds to 
$N_0:=\{ Q: Q:=a_0Q_0+a_1Q_1+a_2Q_2=0 |  \det(a_0A_0+a_1A_1+a_2A_2 )=0\}$
where $A_i$ is the $6\times 6$ symmetric matrix corresponding to the
quadric $Q_i$.
Since $\det(a_0A_0+a_1A_1+a_2A_2 )=0$ is a homogeneous polynomial of degree
6 in the variables $a_0,a_1,a_2$, the locus $N_0$ is a sextic curve in 
$N\cong\PP^2$. 

Since we assumed that every quadric containing $Y$ has rank $\geq 5$, $N_0$
is a smooth sextic curve. For a proof, see \cite{T}, Chapter~2, Section~3.

It turns out that the moduli space $M$ of $\Oh_Y(1)$-semistable sheaves on $Y$
with Mukai vector $(2,\Oh_Y(1),2)$ is isomorphic to the double cover of 
$N\cong \PP^2$, branched along $N_0$ (see \cite{M2}). 

We will explain below the relationship between vector bundles on $Y$ with 
Mukai vector $(2,\Oh_Y(1),2)$ and points on $M$.  Readers familiar with 
the material can skip to \ref{S-const}.




Let $G$ be the Plucker embedding of $G(1,3)$, the Grassmannian of two planes
in $\C^4$ or equivalently lines in $\PP^3$.  It is isomorphic to 
the zero locus of the
quadratic equation
    $2z_0 z_5 - z_1 z_4 + z_2 z_3=0$, 
where $z_i$ are the homogeneous co-ordinates on $\PP^5$. In
$G$ there exist two families of 2-planes corresponding to the Schubert
cycles
$$\begin{array}{llll}
\sigma(p) & = &  \{\ell \subseteq \PP^3 : p \in \ell \} & \text{where $p$ is a
point in $\PP^3$}\\ 
\sigma(h) & = & \{\ell \subseteq \PP^3 : \ell \subseteq h\}
& \text{where $h$ is a hyperplane in $\PP^3$}.
\end{array}
$$

Let $y$ be the point in $G$ corresponding to the line $\ell_y$ in
$\PP^3$. Then $T_y(G) \cap G = \{ \,\text{lines in}\, G \,\text{passing
through}\, 
y\}$ which
is equal to $\bigcup_{p \in \ell_y} \sigma(p)$ = $\bigcup_{\ell \subset
h} \sigma(h)$ (see \cite{GH}).

In fact $T_y(G) \cap G$ is a cone over a smooth quadric surface
$\overline{Q}$ in
$\PP^3$.  Since such a surface has two distinct rulings of lines on it, it
follows that there are two families of 2-planes in $G$ passing through
$y$, each obtained by joining the lines of one ruling with $y$.  We can
find an interpretation of this in terms of Schubert cycles as well. 
The cohomology group $\h^4(G,\Z)$ is generated by the Schubert cycles 
$\sigma(p)$ and $\sigma(h)$. Then using various formulae for calculating
intersections of Schubert cycles,
we get  that for $p_1 \ne p_2$ and 
$h_1 \ne h_2$,  $\intf{\sigma(p_1),\sigma(p_2)}
= 1 = \intf{\sigma(h_1),\sigma(h_2)}$ and for 
for $i=1,2,\, j=1,2,\,\intf{\sigma(p_i),\sigma(h_j)} = 0$ 
(see \cite{GH}). Since 
2-planes of opposite rulings intersect in a line, while those of the same
family intersect at the point $y$, it follows that the 2-planes spanned by
$y$ and the lines of one ruling must be all the Schubert cycles
$\sigma(p)$ for $p \in l_y$ while those spanned by the lines of the other
ruling must be all the cycles $\sigma(h)$ such that $\ell \subset h$.

\begin{lemma} \label{L:pre} 
Let $y$ be a point in a smooth quadric
hypersurface $Q$ in $\PP^5$.  Consider one of the two families of
2-planes contained in the quadric.  Then the set of 2-planes in that
family which pass through $y$ form a smooth conic curve in $G(2,5)$.
\end{lemma} 

\begin{proof} 
We have already shown that on $G$ such a family arises
from the span of $y$ and the lines of a ruling on a smooth quadric surface
$\overline{Q}$.  

Any two quadrics in $\PP^5$ having the same rank are isomorphic via
the action of $\PP GL(6)$. So there exists an element of $\PP GL(6)$ mapping
$Q$ to $G$. Since $\PP GL(6)$ preserves linear subspaces, the element
under consideration maps the families of 2-planes contained in $Q$ to 
families of 2-planes in $G$. So it suffices to consider only $G$. 



Consider now $C = \{\ell \subseteq \PP^3 : \ell \subseteq \overline{Q}\}$.  
This is a
one dimensional variety with two smooth components $C_1, C_2$.  Our family
of 2-planes is isomorphic to one of the two components.  To find the degree
of $C_i$ we take its intersection with a hyperplane class
$\sigma(\ell_0)$ of $G(1,3)$. So $\sigma(\ell_0) \cap \overline{Q} = 
\{\ell \subset \overline{Q} : \ell \cap
\ell_0 \neq \emptyset\}$. A general line $\ell_0$ 
will intersect $\overline{Q}$ in two
distinct points.  One line from each ruling passes through each point,
therefore the degree of each $C_i$ is 2, which proves the lemma.
\end{proof}

\begin{note}
In the case that $Q$ is singular at a point, and $x$ is still a
smooth point, we have that $T_y(Q) \cap Q$ is a cone over a
quadric cone $A$ in $\PP^3$.  In that case there is only one ruling of
lines on $A$ and hence only one family of 2-planes contained in $Q$ and
passing through $y$.  So $C$ as above has only one component.  Again we
take the intersection of $\sigma(\ell_0)$ with $A$, which is equal to 
$\{\ell
\subset A : \ell \cap \ell_0 \neq \emptyset\}$.  A general line $\ell_0$ will
intersect $A$ in 2 points and therefore the degree of $C$ in this case is
also 2.
\end{note}

 Consider the universal sequence on G 
 $$ 0 \ra S \ra \C^4 \ra Q \ra 0.$$ 
 Let $S_y , Q_y$ denote the fibres over $x$ of $S$ and $Q$
respectively. 

\begin{note}
Here $Q$ stands for the universal quotient bundle on the Grassmannian $G$
and not a quadric hypersurface in $\PP^5$. We are using the same notation 
as in \cite{GH}.
\end{note}
 
Consider the incidence correspondence $I_1$ in $Y \times G(2,5)$, where
$$I_1 = \{(y,\Lambda) : y \in \Lambda, \Lambda \in \sigma(h)\ ,\ell_y\subset h 
\}.$$ 
Then $I_1 \ra Y$ is a smooth $\PP^1$ bundle.  Over each
$x$ the fibre is $\{h \subseteq \PP^3 : \ell_y \subset h\}$, which is
isomorphic to $\PP(\C^4/S_y)$, and so $I_1 = \PP(Q)$.  Similarly we define
$$I_2 = \{(y, \Lambda) : \Lambda = \sigma(p), p \in \ell_y\}.$$  
$I_2$ is a
$\PP^1$ bundle where now the fibre over each $y$ is the set of points on
$l_y$, i.e. the set of hyperplanes in $\ell_y$. But this is just 
$\ell^\vee_y = \PP(S^\vee_y)$.  So in fact $I_2 = \PP(S^\vee)$. $S^\vee$ and
$Q$ are rank 2 vector bundles on G which when restricted to $Y$ have Mukai
vector $(2,\Oh(1),2)$.

Since all the smooth quadrics in $\PP^5$ are isomorphic to $G$ we obtain
two families of 2-planes for each quadric. In the case of a singular quadric
(of rank at least 5), 
the two families degenerate into one. Consider the net of quadrics
generated by three smooth quadrics $Q_0, Q_1, Q_2$.  The family of
2-planes sweeping out the quadrics in the net forms a double cover of
$N\cong\PP^2$ branched along a smooth sextic curve $N_0$.  The curve $N_0$ 
represents the
locus of singular quadrics in the net.  Associated to each family of
2-planes in a quadric is a $\PP^1$-bundle which is the projectivization
of a rank 2 vector bundle on $Y$ with Mukai vector $(2,\Oh(1),2)$.  It can be
shown that these represent all the isomorphism classes of such bundles,
(see \cite{N}).  
This shows that $M$ is isomorphic to the double cover of $\PP^2$ branched
along the sextic. In the case when the rank of every quadric containing $Y$ 
is not $\geq 5$, the locus of singular quadrics with rank $\leq 4$ is a
codimension one subset of $N_0$. In that case,
we get a rational map from the double cover 
of $\PP^2$
(branched along the  sextic curve $N_0$) to $M$.  Mukai's result 
(see \cite{M1},
\cite{M2}) 
then shows that
in fact $M$ is isomorphic to the double cover of $N\cong \PP^2$ branched
along $N_0$.


\begin{claim}
The generic $M$ is not a fine moduli space.
\end{claim}

\begin{proof}
This  follows 
from a result by Newstead,(see\cite{N}).  Generally $\Pic(M) \cong \Z\,D$
,
generated by the pull back of $\Oh(1)$ from $\PP^2$. (Recall that $M$ is 
a double cover of $\PP^2$ branched over a sextic). 
Then by adjunction $\intf{D,D} = 2$.  Now suppose $M$ is a fine moduli 
space i.e., 
there exists a rank 2 sheaf F on $Y \times M$ satisfying the desired 
properties.  For each $y \in Y, F_y := F_{|_{Y\times M}}$ is a rank 2 bundle 
with ${c_1}(F) = kD$ for some k.  Newstead shows that for ${F_y}_{|_D}$ such 
that $\PP(F_y)_{|_D} = I_{|_D}$, ${F_y}_{|_D}$ has odd degree.  But that's a 
contradiction 
because degree of ${F_y}_{|_D} = \intf{{c_1}(F_y),D} = k\intf{D,D} = 2k$ which 
is an even number.
\end{proof}




\subsection{The construction} 
\label{S-const} 
We would like
to construct a rank 4 vector 
bundle $V$ on $Y \times M$ such that: $V_{|_{Y\times m}}
\cong (V_1 \oplus V_2)_{|_{Y\times m}}$, and 
$\PP({V_i}_{|_{Y\times m}})$ is isomorphic to
the $\PP^1$ bundle on $X$ associated to the family $m \in M$ of 2-planes
sweeping out a quadric in the net.  (See Definition \ref{defn:QuasiUniv}.) 

 
\begin{defn} 
We consider the incidence variety $I$ in $Y \times M \times G(2,5)$
where $I = \{(y,m,\Lambda) : y \in \Lambda , \Lambda \in m \}$.  Then
$\pi:I \ra Y \times M$ is a $\PP^1$ bundle, where the fibre over each
$(y,m)$ is a smooth conic curve denoted by $C$.  The pull back of $\Oh(1)$
from $G(2,5)$ to $I$ is a line bundle $L$ such that $L_{|_C} \cong
\Oh_{\PP^1}(2)$.
\end{defn}

\begin{lemma} We have the following short exact sequence 
$$0 \ra K \ra {\pi^*}{\pi_*}L \ra
L \ra 0,$$
 where the last map is the evaluation map. 
\end{lemma}

\begin{proof}
It is enough to show this fibrewise, i.e. we need to show that 
the evaluation map 
${\pi^*}{\pi_*}L_{|_C} \cong \Oh_C \tensor \h^0(\Oh(2)) \ra L_{|_C}$ is 
surjective.
If not then there exists a point $p$ in $C$ such that $\sigma_0(p) =
\sigma_1(p) = \sigma_2(p) = 0$ where $\sigma_0, \sigma_1, \sigma_2$
are a basis for $\h^0(\PP^1, \Oh(2))$.  But that implies
$h^0(\PP^1,\Oh(2)) = h^0(\PP^1,\Oh(1))$.  By Riemann Roch we get that
$h^0(\PP^1, \Oh(2)) = 3$ and  $h^0(\PP^1,\Oh(1)) = 2$, so we have a
contradiction.
\end{proof}



Now
$K$ is a rank 2 bundle on $I$ such that $K_{|_C}$ has degree -2.
Since all vector bundles on $\PP^1$ are decomposable we get that
$K_{|_C} \cong \Oh_C(a) \oplus \Oh_C(b)$ where $a + b = -2$

The long exact cohomology sequence associated to the sequence above in
the lemma is as follows:  $0 \ra \h^0(K_{|_C}) \ra \h^0(\PP^1,\Oh(2)) \ra
\h^0(L_{|_C}) \ra \h^1(K_{|_C}) \ra \cdots$

Since the second map is an isomorphism we get that $\h^0(K_{|_C}) = 0$,
which implies that $K_{|_C} \cong {\Oh_{\PP^1}(-1) \oplus \Oh_{\PP^1}(-1)}$.
Hence $K^{\vee}_{|_C} \cong {\Oh_{|_{\PP^1}}(1) \oplus \Oh_{|_{\PP^1}}(1)}.$

Let $V = \pi_*(K^\vee)$.  Then $V$ is a rank 4 vector bundle on $Y
\times M$ such that $V_{| Y\times m} \cong (V_1 \oplus V_2)_{| Y\times m}$ 
where ${V_1}_{| Y\times m} \cong {V_2}_{| Y\times m} $ and
the restriction of $\PP(V_i)$ to $(y,m)$ is isomorphic to $C$ which is
the fibre of $I$ over $(y,m)$.   So $V$ is a quasi-universal sheaf
(up to isomorphism), associated to the moduli space of sheaves on X
with Mukai vector $(2,\Oh(1),2)$.


\subsection{Application}



This example lends itself to a nice application of a result by C\u ald\u araru.

Recall that we had an embedding
$$0 \ra T_Y \ra T_M \ra \Z/n\Z \ra 0$$
where $M$ is the moduli space of sheaves on $Y$ as before.
So $T_Y$ is the kernel of some element 
$\alpha \in T^\vee_M\tensor \Q/\Z= 
\Br(M)$.

\begin{notation}
For $\omega \in \widetilde{\h}(M,\Q)$,
the functional $\intf{\omega,\cdot}_{|_{\tm}} \mod \Z$ as an element
of  
$\Hom(\tm,\Q/\Z)=\Br(M)$ will be denoted by $[\omega]$.
\end{notation}

C\u ald\u araru has proved the following generalization 
of Mukai's result:



\begin{thm}
Let $Y$ be an algebraic K3 surface. Let $v$ be a primitive
isotropic Mukai vector, and let $M$ be the moduli space 
of $\Oh_Y(1)$-semistable sheaves with Mukai vector $v$.
Let $n=\gcd\{\intf{\omega,v}\}$ where $\omega$ runs over all elements of 
$\widetilde\h^{1,1}(Y,\Z)$. Assume that $M$ 
is compact and nonempty (so $M$ is a K3 surface) and 
by Mukai's results we have
the map
$Z_\E: \widetilde\h(Y,\Q) \ra \widetilde\h(M,\Q)$.
If $u\in \widetilde\h(Y,\Z)$ such that $\intf{u,v}=1\pmod n$
then
\begin{enumerate}
\item
$\alpha:=[Z_\E(u)]\in \Br(M)$ has $n$ torsion and is the obstruction to 
the existence of a universal sheaf on $Y\times M$,
\item
The derived category of sheaves on $Y$, $\D(Y)$, is equivalent
to the derived category of $\alpha$-twisted sheaves on 
$M$, $\D(M,\alpha)$, i.e.
$$\D(Y)\cong\D(M,\alpha).$$
\end{enumerate}

\end{thm}

\begin{note}
If $n=1$, we get that the moduli space is fine. 
In our case, since $v=(2,\Oh(1),2)$ and since for any 
$\omega=(r,\Oh(k),s)$ in $\tilde\h^{1,1}(Y,\Z)$, 
$\intf{v,\omega}=8k-2(r+s)$, we get that $n=2$.
This is another explanation for why the moduli space
in general is not fine.
\end{note}


We can find an explicit representative of the obstruction
class $u$, by moving away from the generic element
in the family $\Y$, to one where $\rank \Pic(Y) \geq 2$.

\begin{thm}
Let $Y$ be a degree 8 surface in $\PP^5$, such that either $Y$ 
contains a line $L$, or $Y$ is an element of $\Mb$. Then
the moduli space associated to the Mukai vector 
$(2,\Oh_Y(1),2)$ is actually a fine moduli space.
\end{thm}

\begin{proof}
We first consider the case when $Y$ contains a line $L$.
The set of all such surfaces forms a codimension one 
locus in the family of all degree $8$ surfaces in $\PP^5$.
Let $u:=(0,L,0)$, then $\intf{u,v}=1$ so the moduli space
is fine.

A more geometric way to see this is the following:

$I \ra Y\times M$ lifts to a rank 2 vector bundle $F$ if and only
if there exists a section $\sigma: Y\times M \ra I$. 



Recall that in terms of Schubert cycles, the hyperplane class,
$H=\Oh_Y(1)$ corresponds to $\sigma(\ell_0)=\{\ell_y: \ell_y\cap \ell_0
\neq \emptyset \}$ where  $\ell_0$ is a general line in $\PP^3$ and
$\ell_y$ is the line corresponding to $y\in Y$ (see earlier section).

The line $L\subset Y$ corresponds to the Schubert cycle $\sigma(p_0,
h_0)=\{\ell: p_0 \in \ell \,\text{and}\, \ell\subset h_0 \}$ for some
point $p_0$ and some 2-plane $h_0$ containing $p_0$ in $\PP^3$.

$L\cdot H=1$ implies that for a general $y\in Y$, there exists exactly
one line $\ell_{L,y}$ in $\sigma(p_0,h_0)$ such that $\ell_{L,y}\cap
\ell_y \neq \emptyset $.


Consider the fibre $C=I_{y,m}$. Then either 
\begin{enumerate}
\item 
$C \cong \{ \sigma(p): p \in \ell_y \}$ or
\item 
$C \cong \{ \sigma(h): \ell_y \subset h\}$.
\end{enumerate}
We will define a section of $\pi: I \ra Y\times M$, using $L$.

In the first case, let $\sigma(y,m)$ be the unique point in $C$ corresponding
to the 2-plane $\sigma(p')$ where $p'=\ell_y\cap \ell_{L,y}$.



In the second case, map $(y,m)$ to the point in $C$, corresponding to 
\emph{unique 2-plane} $h_0$, spanned by $\ell_y$ and $\ell_{L,y}$.

This defines a section $\sigma: Y\times M \ra  I$. 
 It follows that $I \cong \PP( \pi_*(\Oh(\sigma)^\vee))$ and
so $\pi_*(\Oh(\sigma)^\vee)$ is a universal sheaf.

This is in agreement with C\u ald\u araru's result. As before, let $L$ denote
the class of the line and let $u=(0,L,0)$, then $\intf{u,v}=1$, so 
$gcd \{\intf{u,v}\} = 1$.
Then for any $u\in \widetilde\h(Y,\Z)$ such that $\intf{u,v}=1$, 
$Z_\E(u)\in \widetilde\h(M,\Z) $, hence $[Z_\E(u)]\in \Br(M)$ is trivial.
By C\u ald\u araru's Theorem, $[Z_\E(u)]$ corresponds to the obstruction, which
in this case is trivial, hence $M$ is a fine moduli space.



Next we consider the case when $Y$ is an element of the family $\Mb$,
i.e. $Y$ is a genus one fibration such that $\intf{F,H}=3$ where $F$ is
the fibre class and $H$ is the hyperplane class. 

Let $u:=(0,F,0)$ then 
$$\intf{(0,F,0),(2,\Oh(1),2)}=3\equiv 1 \pmod 2,$$ 
so $\gcd\{\intf{u,v}\}=1$ where $u$ runs over elements of
$\widetilde\h^{1,1}(Y,\Z)$, which  as before implies that $M$ is a fine
moduli space.  
\end{proof}

Consider a marked deformation of $Y$ in $\Y$ so that the class
$F$ becomes transcendental. Let $\alpha=[Z_\E(F)]$. Then by C\u ald\u araru's
Theorem, (see \cite{C1})
$$ \D(Y) \cong  \D(M,\alpha). $$

It is hoped that the above construction of the quasi-universal
sheaf will lead to a better insight of C\u ald\u araru's conjecture for other
non fibred algebraic K3 surfaces.






\begin{thebibliography}{BKR}


\bibitem[BPV]{BW} W.~Barth, C.~Peters and A.Van de Ven, \emph{Compact
Complex Surfaces}, Springer-Verlag, Berlin (1984).


\bibitem[C]{C1} A.~C\u ald\u araru, \emph{Non-fine moduli spaces of
sheaves on K3 surfaces}, Int.Maths.Res.Not{\bf20} 1027-1056 (2002).
(also preprint math.AG/\linebreak[2]0108180).


\bibitem[CDFKM]{Cetal} P.~Candelas, X.~de la Ossa, A.~Font, S.~Katz and
D.~Morrison, \emph{Mirror symmetry for two parameter models I},
Nucl. Phys. B{\bf419} (1994) 352--403, hep-th/9308083.


\bibitem[DG]{DG} I.~Dolgachev, M.~Gross, \emph{Elliptic Threefolds I: 
Ogg-Shafarevich Theory}, J. Alg. Geom 3 (1994), 39--80 (also 
preprint, alg-geom/9210009).

\bibitem[DP]{DP} R.~Donagi and T.~Pantev, \emph{Torus fibrations,
gerbes, and duality}, preprint math.AG/\linebreak[2]0306123.


\bibitem[FM]{FM} R.~Friedman, J.~Morgan, \emph{Smooth Four Manifolds and
Complex Surfaces}, Springer-Verlag, Berlin (1994).


\bibitem[G]{G} J.~Giraud, \emph{Cohomologie Non-ab\'elienne}, Grundlehren
Vol. 179, Springer-Verlag, Berlin (1971).


\bibitem[GH]{GH} P.~Griffiths and J.~Harris, \emph{Principles of
Algebraic Geometry}, John Wiley and Sons, (1978).



\bibitem[GS]{GS} M.~Gross and B.~Siebert, \emph{Mirror symmetry via
logarithmic degenerations data 1}, preprint math.AG/\linebreak[2]0309070.


\bibitem[K]{Kh} M.~Khalid, \emph{K3 correspondences}, Ph.D. thesis, 
University of Pennsylvania, July 2003.


\bibitem[L]{L} S.~Lang, \emph{Algebra}, Addison-Wesley, Reading, Mass. (1965).

\bibitem[M1]{M1} S.~Mukai, \emph{On the moduli space of bundles on K3
surfaces I}, M.F. Atiyah and others, Vector bundles on algebraic
varieties, Bombay colloquium 1984, Tata Institute for fundamental
research studies in mathematics \textbf{11} 139--174 (1987).

\bibitem[M2] {M2} S.~Mukai, \emph{Moduli of vector bundles on K3
surfaces and, and symplectic manifolds}, sugaku Expositions, 
\textbf{1} n.2, 139-174(1988).


\bibitem[M3]{M3} S.~Mukai, \emph{Vector bundles on a K3 Surface}, arXiv:
math.AG/\linebreak[2]0304303


\bibitem[N]{N} P.E.~Newstead, \emph{Stable Bundles of Rank 2 and Odd
Degree over a Curve of Genus 2}, Topology \textbf{7} 205--215 (1968).

\bibitem[O]{O} D.~Orlov, \emph{Equivalences of Derived Categories and 
K3 Surfaces}, preprint alg-geom/9606006.


\bibitem[SYZ]{SYZ} A.~Strominger, S.T. Yau and E.~Zaslow,
\emph{Mirror symmetry is T-duality}, Nucl. Phys B{\bf479} (1996)
243--259, also hep-th/9606040.

\bibitem[T]{T} A.~N.~Tyurin, \emph{Intersection of Quadrics}, Russian
Math. Surveys \textbf{30:6}, 51-105 (1975).


%\bibitem[F]{RF} R.Friedman, \emph{Algebraic Surfaces and Holomorphic Vector
%Bundles}, Springer Verlag, New York (1998).








%\bibitem[K]{K} S.~Katz, \emph{Rational Curves on Calabi-Yau Threefolds}, 
%preprint alg-geom/9312009.




%\bibitem[Ma]{M} A.~Mayers, \emph{Families of K3 Surfaces},  Nagoya.J.Math 
%\textbf{48} (1972) 1-17.

%\bibitem[Mo]{Mo} D.~Morrison, \emph{Geometric Aspects of Mirror Symmetry}, 
%Mathematics Unlimited--2001 and beyond, Springer-Verlag, 2001, pp 899-918.







%\bibitem[RT]{RT} F.~Rodriguez-Villega, J.~Tate., \emph{Jacobian of Plane 
%Cubics}, at www.ma.utexas.edu/users/villegas/cnt/jac\_quart.gp.



%\bibitem[TY]{TY} R.~Thomas, S.~Yau, \emph{Special Lagrangians, Stable 
%Bundles and Mean Curvature Flow}, preprint math.DG/0104197.

%\bibitem[VW]{VW} K.~Vafa, E.~Witten, \emph{On Orbifolds with Discrete Torsion}
%, J.Geom.Phys \textbf{15} 189-214 (1995).

%\bibitem[W]{W} A.~Weil, \emph{Euler and the Jacobians of Elliptic Curves},  
%Arithmetic and Geometry, Birkhauser Boston Inc., Boston, \textbf{1} 353-359
%(1983).



\end{thebibliography}

\end{document}


math0402314.extracted_bib
\begin{thebibliography}{BKR}


\bibitem[BPV]{BW} W.~Barth, C.~Peters and A.Van de Ven, \emph{Compact
Complex Surfaces}, Springer-Verlag, Berlin (1984).


\bibitem[C]{C1} A.~C\u ald\u araru, \emph{Non-fine moduli spaces of
sheaves on K3 surfaces}, Int.Maths.Res.Not{\bf20} 1027-1056 (2002).
(also preprint math.AG/\linebreak[2]0108180).


\bibitem[CDFKM]{Cetal} P.~Candelas, X.~de la Ossa, A.~Font, S.~Katz and
D.~Morrison, \emph{Mirror symmetry for two parameter models I},
Nucl. Phys. B{\bf419} (1994) 352--403, hep-th/9308083.


\bibitem[DG]{DG} I.~Dolgachev, M.~Gross, \emph{Elliptic Threefolds I: 
Ogg-Shafarevich Theory}, J. Alg. Geom 3 (1994), 39--80 (also 
preprint, alg-geom/9210009).

\bibitem[DP]{DP} R.~Donagi and T.~Pantev, \emph{Torus fibrations,
gerbes, and duality}, preprint math.AG/\linebreak[2]0306123.


\bibitem[FM]{FM} R.~Friedman, J.~Morgan, \emph{Smooth Four Manifolds and
Complex Surfaces}, Springer-Verlag, Berlin (1994).


\bibitem[G]{G} J.~Giraud, \emph{Cohomologie Non-ab\'elienne}, Grundlehren
Vol. 179, Springer-Verlag, Berlin (1971).


\bibitem[GH]{GH} P.~Griffiths and J.~Harris, \emph{Principles of
Algebraic Geometry}, John Wiley and Sons, (1978).



\bibitem[GS]{GS} M.~Gross and B.~Siebert, \emph{Mirror symmetry via
logarithmic degenerations data 1}, preprint math.AG/\linebreak[2]0309070.


\bibitem[K]{Kh} M.~Khalid, \emph{K3 correspondences}, Ph.D. thesis, 
University of Pennsylvania, July 2003.


\bibitem[L]{L} S.~Lang, \emph{Algebra}, Addison-Wesley, Reading, Mass. (1965).

\bibitem[M1]{M1} S.~Mukai, \emph{On the moduli space of bundles on K3
surfaces I}, M.F. Atiyah and others, Vector bundles on algebraic
varieties, Bombay colloquium 1984, Tata Institute for fundamental
research studies in mathematics \textbf{11} 139--174 (1987).

\bibitem[M2] {M2} S.~Mukai, \emph{Moduli of vector bundles on K3
surfaces and, and symplectic manifolds}, sugaku Expositions, 
\textbf{1} n.2, 139-174(1988).


\bibitem[M3]{M3} S.~Mukai, \emph{Vector bundles on a K3 Surface}, arXiv:
math.AG/\linebreak[2]0304303


\bibitem[N]{N} P.E.~Newstead, \emph{Stable Bundles of Rank 2 and Odd
Degree over a Curve of Genus 2}, Topology \textbf{7} 205--215 (1968).

\bibitem[O]{O} D.~Orlov, \emph{Equivalences of Derived Categories and 
K3 Surfaces}, preprint alg-geom/9606006.


\bibitem[SYZ]{SYZ} A.~Strominger, S.T. Yau and E.~Zaslow,
\emph{Mirror symmetry is T-duality}, Nucl. Phys B{\bf479} (1996)
243--259, also hep-th/9606040.

\bibitem[T]{T} A.~N.~Tyurin, \emph{Intersection of Quadrics}, Russian
Math. Surveys \textbf{30:6}, 51-105 (1975).


%\bibitem[F]{RF} R.Friedman, \emph{Algebraic Surfaces and Holomorphic Vector
%Bundles}, Springer Verlag, New York (1998).








%\bibitem[K]{K} S.~Katz, \emph{Rational Curves on Calabi-Yau Threefolds}, 
%preprint alg-geom/9312009.




%\bibitem[Ma]{M} A.~Mayers, \emph{Families of K3 Surfaces},  Nagoya.J.Math 
%\textbf{48} (1972) 1-17.

%\bibitem[Mo]{Mo} D.~Morrison, \emph{Geometric Aspects of Mirror Symmetry}, 
%Mathematics Unlimited--2001 and beyond, Springer-Verlag, 2001, pp 899-918.







%\bibitem[RT]{RT} F.~Rodriguez-Villega, J.~Tate., \emph{Jacobian of Plane 
%Cubics}, at www.ma.utexas.edu/users/villegas/cnt/jac\_quart.gp.



%\bibitem[TY]{TY} R.~Thomas, S.~Yau, \emph{Special Lagrangians, Stable 
%Bundles and Mean Curvature Flow}, preprint math.DG/0104197.

%\bibitem[VW]{VW} K.~Vafa, E.~Witten, \emph{On Orbifolds with Discrete Torsion}
%, J.Geom.Phys \textbf{15} 189-214 (1995).

%\bibitem[W]{W} A.~Weil, \emph{Euler and the Jacobians of Elliptic Curves},  
%Arithmetic and Geometry, Birkhauser Boston Inc., Boston, \textbf{1} 353-359
%(1983).



\end{thebibliography}