Manifolds covered by lines, defective manifolds\\ and a restricted Hartshorne Conjecture
Paltin Ionescu*
Small codimensional embedded manifolds defined by equations of small degree are Fano and covered by lines. They are complete intersections exactly when the variety of lines through a general point is so and has the right codimension. This allows us to prove the Hartshorne Conjecture for manifolds defined by quadratic equations and to obtain the list of such Hartshorne manifolds. Using the geometry of the variety of lines through a general point, we characterize scrolls among dual defective manifolds. This leads to an optimal bound for the dual defect, which improves results due to Ein. We discuss our conjecture that every dual defective manifold with cyclic Picard group should also be secant defective, of a very special type, namely a local quadratic entry locus variety.
0909.2763.tex
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\begin{document}

\begin{abstract}Small codimensional embedded manifolds defined by equations of small degree are Fano and covered by lines. They are complete intersections exactly when the variety of lines through a general point is so and has the right codimension. This allows us to prove the Hartshorne Conjecture for manifolds defined by quadratic equations and to obtain the list of such Hartshorne manifolds. Using the geometry of the variety of lines through a general point, we characterize scrolls among dual defective manifolds. This leads to an optimal bound for the dual defect, which improves results due to Ein. We discuss our conjecture that every dual defective manifold with cyclic Picard group should also be secant defective, of a very special type, namely a local quadratic entry locus variety.
\end{abstract}
\subjclass[2000]{14MXX, 14NXX, 14J45, 14M07}
\keywords{Fano manifold, covered by lines, dual and secant defective, scroll, quadratic manifold, Hartshorne Conjecture}
\title[Manifolds covered by lines]{Manifolds covered by lines, defective manifolds\\ and a restricted Hartshorne Conjecture}
\author[Paltin Ionescu]{Paltin Ionescu*}
\address{\sc University of Bucharest, Faculty of Mathematics and Computer Science,\linebreak 14 Academiei St., 010014 Bucharest\newline \noindent and \newline\indent Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest\\ Romania}
\email{Paltin.Ionescu@imar.ro}
\author[Francesco Russo]{Francesco Russo}
\address{\sc Dipartimento di Matematica e Informatica\\
Universit\` a degli Studi di Catania\\
Viale A. Doria, 6\\
95125 Catania\\ Italy}
\email{frusso@dmi.unict.it}


\thanks{*Partially  supported  by the Italian Programme ``Incentivazione alla mobilit\`{a} di studiosi stranieri e italiani residenti all'estero"}
\maketitle

\section*{Introduction}

The present paper is a natural sequel to our previous work, see \cite{Ru, IR, IR2} and also \cite{BI}. The geometry of the variety of lines passing through the general point of an embedded projective manifold is investigated, in the framework of Mori Theory. Two, rather different, applications are obtained. First, scrolls are characterized among all dual defective manifolds, and the latter are related to some special secant defective ones. Secondly, the quadratic case of the Hartshorne Conjecture is proved, and all extremal examples other than complete intersections are described.

We consider $n$-dimensional irreducible non-degenerate complex projective manifolds $X\subset \p^{n+c}$. We call $X$ a {\it prime Fano manifold of index $i(X)$} if its Picard group is generated by the hyperplane section class $H$ and $-K_X=i(X)H$ for some positive integer $i(X)$. One consequence of Mori's work \cite{Mori} is that, for $i(X) > \frac{n+1}{2}$, $X$ is {\it covered by lines}, i.e.\ through each point of $X$ there passes a line, contained in $X$. As noticed classically, Fano {\it complete intersections} with $i(X)\geq 2$ are also covered by lines. The ``biregular part'' of Mori Theory
(no singularities, no flips, ...), see \cite{Mori2, De, Ko}, provides the natural setting for studying manifolds covered by lines. For instance, as first noticed in \cite{BSW}, when the dimension of the variety of lines passing through a general point is at least $\frac{n-1}{2}$, there is a Mori contraction of the covering family of lines. Moreover, its general fiber (which is still covered by lines) has cyclic Picard group, thus being a prime Fano manifold. For prime Fanos, the study of covering families of lines is nothing but the classical aspect in the theory of the {\it variety of minimal rational tangents}, developed by Hwang and Mok in a remarkable series of papers, see e.g. \cite{HM, HM2, HM3, Hwang}. We also recall that lines contained in $X$ play a key role in the proof of an important  result due to Barth--Van de Ven and Hartshorne, cf. \cite{Barth}. It states that $X$ must be a complete intersection when its dimension is greater than a suitable (quadratic) function of its degree.

Prime Fanos of high index other than complete intersections are quite rare. For instance, {\it dual defective} manifolds and some special but important {\it secant defective} ones provide such examples. Thus, when ${\rm Pic}(X)=\mathbb Z\langle H\rangle$, the class of {\it local quadratic entry locus} varieties, see \cite{Ru, IR, IR2} gives examples with $i(X)=\frac{n+\delta}{2}$, $\delta$ being the secant defect, while dual defective manifolds have $i(X)=\frac{n+k+2}{2}$, where $k$ is the dual defect, see \cite{Ein}.

Another intriguing fact is that all known examples of prime Fanos of high index are either complete intersections or {\it quadratic}, i.e. scheme theoretically defined by quadratic equations. Mumford, \cite{Mum}, was the first to call the attention to the fact that many special but highly interesting embedded manifolds are quadratic. One crucial remark we made is that prime Fanos of high index are embedded with {\it small codimension}, which naturally leads us to the famous {\it Hartshorne Conjecture}, see~\cite{Ha}:
if $n\geq 2c+1$, $X$ should be a complete intersection.
This conjecture is already very difficult even in the special case of (prime) Fano manifolds. Early contributions related to the conjecture came by with a topological flavor. First the celebrated Fulton--Hansen Connectivity Theorem \cite{FH}, followed by Zak's Linear Normality Theorem and his equally famous Theorem on Tangencies, see \cite{Zak}. Then, the beautiful result of Faltings \cite{Faltings}, later improved by Netsvetaev \cite{Net}, showing that $X$ is a complete intersection when the {\it number} of equations scheme theoretically defining it is small. Another development, suggested by classical work of Severi, is due to Bertram--Ein--Lazarsfeld \cite{BEL}. They characterize complete intersections in terms of the {\it degrees} of the first $c$ equations defining $X$ (in decreasing order).

Consider now $x\in X$ a general point and let $\mathcal{L}_x$ denote the variety of lines through $x$, contained in $X$. Note that $\mathcal{L}_x$ is naturally embedded in $\p^{n-1}=$ the space of tangent directions at $x$. Let $a:=\dim(\mathcal{L}_x)$ and note that for prime Fanos, $i(X)=a+2$. When $a\geq \frac{n-1}{2}$, $\mathcal{L}_x\subset \p^{n-1}$ is smooth, irreducible and, due to a key result by Hwang \cite{Hwang}, non-degenerate. Our Theorem~\ref{schemeLxrefined}, inspired by \cite[Corollary~4]{BEL}, shows that we can relate the equations of $\mathcal{L}_x\subset \p^{n-1}$ to those of $X\subset \p^N$. In particular, manifolds of small codimension and defined by equations of small degree are covered by lines; moreover, $X\subset \p^N$ is a complete intersection if and only if $\mathcal{L}_x\subset \p^{n-1}$ is so and its codimension is the right one. In case $X$ is quadratic, we combine the above with Faltings' Criterion \cite{Faltings} to deduce that, when $n\geq 2c+1$, $\mathcal{L}_x\subset \p^{n-1}$ is a complete intersection.  Next, we appeal to a different ingredient, leading to a proof of the Hartshorne Conjecture for quadratic manifolds, see Theorem~\ref{HaCo}. The necessary new piece of information is provided by the projective second fundamental form which, due to the Fulton--Hansen Connectivity Theorem, turns out to be of maximal dimension. We would like to point out that, working with local differential geometric methods, in the spirit of \cite{GH, IL}, Landsberg in \cite{Landsberg} proved that when $n\geq 3c+b-1$ a (possibly singular) quadratic variety $X$ is a complete intersection, where $b=\dim({\rm Sing}(X))$.

Using Netsvetaev's Theorem \cite{Net}, we show that the only quadratic manifolds $X\subset \p^\frac{3n}{2}$ which are not complete intersections are $\mathbb G(1,4)\subset\p^9$ and the spinorial manifold $S^{10}\subset\p^{15}$, see our Theorem~\ref{HV}.

In a somehow opposite direction, starting from an observation due to Buch \cite{Bu}, we show that for a manifold covered by lines, when $a\geq n-c$ each line from the covering family is part of the contact locus of a suitable hyperplane. This property never holds for complete intersections. We elaborate on this and prove that under some mild condition (which is, conjecturally, automatically fulfilled) for dual defective manifolds with cyclic Picard group each general line is a contact line. This allows us, using also \cite{BFS}, to characterize scrolls among all dual defective manifolds, see Theorem~\ref{scrollimit}.
As a consequence, we get an optimal bound on the dual defect, generalizing one of the main results in Ein's foundational papers \cite{Ein, Ein2}. His other main result is recovered, with a different proof. We also give evidence for our conjecture asserting that dual defective manifolds with cyclic Picard group should be local quadratic entry locus varieties. 





\section{Preliminaries}
\begin{enumerate}
\item[$(*)$] {\it Setting, terminology and notation}
\end{enumerate}


Throughout the paper we consider $X\subset \p^N$ an irreducible complex projective manifold
of dimension $n\geq 1$. $X$ is assumed to be non-degenerate and $c$ denotes its codimension, so that $N=n+c$. We also suppose that $X$ is scheme theoretically an intersection of $m$ hypersurfaces of degrees $d_1\geq d_2\geq\cdots\geq d_m$. It is implicitly assumed that $m$ is minimal, i.e. none of the hypersurfaces contains the intersection of the others. We put
$d:=\sum_{i=1}^{c}(d_i-1)$. $X\subset\p^N$ is called {\it quadratic} if it is scheme theoretically an intersection of quadrics (i.e. $d_1=2$). Note that this happens precisely when $d=c$.

For $x\in X$ we let $\mathbf{T}_xX$ denote the (affine) Zariski tangent space to $X$ at $x$, and write $T_xX$ for its projective closure in $\p^N$.
$H$ denotes a hyperplane section (class) of $X$. As usual, $K_X$ stands for the canonical class of $X$. Also, if $Y\subset X$ is a submanifold, we denote by $N_{Y/X}$ its normal bundle. For a vector bundle $E$, $\p(E)$ stands for its projectivized bundle, using Grothendieck's convention.

We let $SX\subset \p^N$ be the {\it secant variety} of $X$, that is the closure of the locus of secant lines. The {\it secant defect} of $X$ is the (nonnegative) number $\delta=\delta(X):=2n+1-\dim(SX)$. We say $X$ is {\it secant defective} when $\delta > 0$.

A secant defective manifold $X\subset \p^N$ is called a {\it local quadratic entry locus} manifold, LQELM for short, if any two general points $x, x' \in X$ belong to a $\delta$-dimensional quadric  $Q_{x,x'}\subset X$. See \cite{Ru, IR2} for a systematic study of these special secant defective manifolds.

$X\subset \p^N$ is {\it conic-connected} if any two general points $x, x'\in X$ are contained in some conic $C_{x,x'}\subset X$. Clearly, any LQELM is also conic-connected. Classification results for conic-connected manifolds are obtained in \cite{IR}, working in the general setting of rationally connected manifolds.

For a general point $x\in X$, we denote by $\mathcal L_x$ the (possibly empty) scheme of lines contained in $X$ and passing through $x$. We say that $X\subset \p^N$ is {\it covered by lines} if $\mathcal L_x$ is not empty for $x\in X$ a general point. We refer the reader to \cite{De, Ko} for standard useful facts about the deformation theory of rational curves; we shall use them implicitly in the simplest case, that is lines on $X$.

Recall that $X$ is Fano if $-K_X$ is ample. The {\it index} of $X$, denoted by $i(X)$, is the largest integer $j$ such that $-K_X = jA$ for some ample divisor $A$.


\begin{enumerate}
\item[$(**)$] {\it The projective second fundamental form}
\end{enumerate}


We recall some general results which 
are probably well known to the experts but for which we are unable to provide a proper reference.

 There are several possible equivalent
definitions of the projective second fundamental form
$|II_{x,X}|\subseteq\p(S^2(\mathbf{T}_xX))$ of an irreducible
projective variety $X\subset\p^N$ at a general point $x\in X$, see
for example \cite[3.2 and end of Section~3.5]{IL}. We use the
one related to tangential projections, as in \cite[Remark
3.2.11]{IL}.

 Suppose $X\subset\p^N$ is non-degenerate, as always,
 let $x\in X$ be a general point and
consider the projection from $T_xX$ onto a disjoint $\p^{c-1}$
\begin{equation}\label{tangentdef}
\pi_x:X\map W_x\subseteq\p^{c-1}.
\end{equation}
The map $\pi_x$ is associated to the linear system of hyperplane
sections cut out by hyperplanes containing $T_xX$, or equivalently
by the hyperplane sections singular~at~$x$.

Let $\phi:\Bl_xX\to X$ be the blow-up of $X$ at $x$, let
\[E=\p((\mathbf{T}_xX)^*)=\p^{n-1}\subset\Bl_xX\] be the exceptional
divisor and let $H$ be a hyperplane section of $X\subset\p^N$. The
induced rational map $\widetilde{\pi}_x:\Bl_xX\map\p^{c-1}$ is
defined as a rational map along $E$ since $X\subset\p^N$ is not a linear space; see
for example the argument in \cite[2.1~(a)]{Ein}. The restriction of
$\widetilde{\pi}_x$ to $E$ is given by a linear system in
$|\phi^*(H)-2E|_{|E}\subseteq|-2E_{|E}|=|\O_{\p((\mathbf{T}_xX)^*)}(2)|=\p(S^2(\mathbf{T}_xX))$.
 

\begin{Definition}
 The {\it second fundamental form}
$|II_{x,X}|\subseteq\p(S^2(\mathbf{T}_xX))$ of an irreducible
non-degenerate variety $X\subset\p^N$ of dimension $n\geq 2$ at a
general point $x\in X$ is the non-empty linear system of quadric
hypersurfaces in $\p((\mathbf{T}_xX)^*)$ defining the restriction of
$\widetilde{\pi}_x$ to $E$.
\end{Definition}
 
Clearly  $\dim(|II_{x,X}|)\leq c-1$ and
$\widetilde{\pi}_x(E)\subseteq W_x\subseteq\p^{c-1}$. From this
point of view the base locus on $E$ of the second fundamental form
$|II_{x,X}|$ consists of {\it asymptotic directions}, i.e.\ of
directions associated to lines having a contact of order at least three with
$X$ at $x$. For example, as we shall see, when $X\subset\p^N$ is quadratic,  the base locus of the second fundamental form
consists of points representing tangent lines contained in $X$ and passing
through $x$, so that it is exactly (even scheme theoretically) $\mathcal L_x$. 


\begin{Proposition}\label{secondform} Let $X\subset\p^N$ be
a smooth irreducible non-degenerate variety of secant defect
$\delta\geq 1$.
Then $\dim(|II_{x,X}|)=c-1$ for $x\in X$ a general point.
\end{Proposition}
\begin{proof}
Let notation be as above. It is sufficient to show
that $\dim(\widetilde{\pi}_x(E))=n-\delta$ because
$\widetilde{\pi}_x(E)\subseteq W_x$ and 
$W_x\subseteq\p^{c-1}$ is a non-degenerate variety, whose  dimension
is  $n-\delta$ by the Terracini Lemma.

Let $TX=\bigcup_{x\in X}T_xX$ be the tangential variety of $X$. The
following formula holds
\begin{equation}\label{dimension}
\dim(TX)=n+1+\dim(\widetilde{\pi}_x(E)),
\end{equation}
 see \cite{Terracini} (or
\cite[5.6, 5.7]{GH}  and \cite[Proposition 3.13.3]{IL} for a modern reference).

The variety $X\subset\p^N$ is smooth and secant defective, so that
$TX=SX$ by a theorem of Fulton and Hansen, \cite{FH}. Therefore
$\dim(TX)=2n+1-\delta$ and from \eqref{dimension} we get
$\dim(\widetilde{\pi}_x(E))=n-\delta$, as claimed.
\end{proof}


\begin{enumerate}
\item[$(*\!*\!*)$] {\it Some standard exact sequences}
\end{enumerate}


Let $V$ be a complex vector space of dimension $N+1$ such that $\p(V)=\p^N$.
Consider the restriction of the Euler sequence on $\p^N$ to $X$
\begin{equation}\label{Euler}
0\to \Omega^1_{\p^N|X}\to V\otimes \O_X(-1)\to \O_X\to 0,
\end{equation}
and the exact sequence on $X$
\begin{equation}\label{tangent}
0\to N_{X/\p^N}^*\to  \Omega^1_{\p^N|X}\to \Omega^1_X\to 0.
\end{equation}
From these exact sequences we deduce
\begin{equation}\label{princ}
0\to N_{X/\p^N}^*(1)\to V\otimes \O_X\to \mathcal P_X\to 0,
\end{equation}
and
\begin{equation}\label{E}
0\to \Omega^1_X(1)\to \mathcal P_X\to \O_X(1)\to 0,
\end{equation}
where $\mathcal P_X$ is the first jet bundle of $\O_X(1)$.

If $\mathbb G(n,N)$ denotes the Grassmannian of $n$-planes in $\p^N$, let $\gamma_X:X\to \mathbb G(n,N)$ be the Gauss map of $X$, associating to a point $x\in X$ the point of $\mathbb G(n,N)$
corresponding to the projective tangent space $T_xX$ to $X$ at $x$. By Zak' s Theorem on Tangencies, the morphism $\gamma_X$ is
finite and birational, see \cite[I.2.8]{Zak}.


Let $\mathcal U$ be the universal quotient bundle  on $\mathbb G(n,N)$, which is a locally free sheaf of rank $n+1$. We have a natural surjection
$$V\otimes \O_{\mathbb G(n,N)}\to \mathcal U,$$ inducing the  surjection $$V\otimes\O_X\to \gamma_X^*(\mathcal U).$$ 
Then it is easy to see that 
$\mathcal P_X=\gamma_X^*(\mathcal U)$; so, for every closed point $x\in X$ we have $\p(\mathcal P_X\otimes k(x))=T_xX\subset\p(V)$.

The above surjection gives an embedding, over $X$, of $\p(\mathcal P_X)\to X$ into $\p(V)\times X\to X$ in such a way that the restriction $\pi_1$ of the projection $\p^N\times X\to \p^N$
to $\p(\mathcal P_X)$ maps $\p(\mathcal P_X)$ onto $TX=\bigcup_{x\in X}T_xX$. 

By \eqref{E} we get 
$$\gamma_X^*(\det(\mathcal U))=\det(\mathcal P_X)\simeq \omega_X\otimes\O_X(n+1).$$
 








 


\section{Manifolds covered by lines}



Let $X\subset \p^N$ be as in ($*$). The following examples of manifolds covered by lines are of relevance to us.

\begin{Examples}\label{ex}

\begin{enumerate}
\item $X\subset \p^N$ a Fano complete intersection with  $i(X)\geq 2$. 

\item (Mori) $X\subset \mathbb{P}^N$ a Fano manifold with ${\rm Pic}(X)=\mathbb{Z}\langle H\rangle$ and  $i(X)> \frac {n+1}2$.

\item $X\subset \p^N$ a conic-connected manifold, different from the Veronese variety $v_2(\p^n)$ or one of its isomorphic projections.
%\ro[Cosa vuoi dire esattamente con questa
%frase? Mi sfugge un po' il senso]


\end{enumerate}
\end{Examples}


Fix some irreducible component, say $\mathcal F$, of the Hilbert scheme of lines on $X\subset \p^N$, such that $X$ is covered by the lines in $\mathcal F$. Put $a=: \deg (N_{\ell/X})$ where $[\ell] \in \mathcal F$. Note that
$a\geq 0$ and $a= \dim (\mathcal{F} _x)$, where $x\in X$ is a general point and     $\mathcal{F}_x=\{[\ell] \in\mathcal F\mid x\in \ell\}$. Moreover, we may view $\mathcal{F}_x$ as a closed subscheme of $\p((\mathbf T_xX)^*)\cong \p^{n-1}$.

When the dimension of $\mathcal{F}_x$ is large, the study of manifolds covered by lines is greatly simplified by the following two facts:

First, we may reduce, via a Mori contraction, to the case where the Picard group is cyclic; this is due to Beltrametti--Sommese--Wi\'sniewski, see \cite{BSW}.
Secondly, the variety $\mathcal{F}_x\subset \p^{n-1}$
inherits many of the good properties of $X\subset \p^N$; this is due to Hwang, see \cite{Hwang}. See \cite{BI} for an application of these principles.
\begin{Theorem}\label{reduction}
Assume $a\geq \frac {n-1}2$. Then the following results hold:
\begin{enumerate}
\item {\rm (\cite{BSW})} There is a Mori contraction, say ${\rm cont}_{\mathcal F} : X\to W$, of the lines from $\mathcal F$; let $F$ denote a general fiber of ${\rm cont}_{\mathcal F}$ and let $f$ be its dimension;


\item {\rm (\cite{Wisniewski})} ${\rm Pic}(F) = \mathbb{Z} \langle H_F\rangle$, $i(F)= a+2$ and $F$ is covered by the lines from $\mathcal F$ contained in $F$;

\item {\rm (\cite{Hwang})} $\mathcal {F}_x\subseteq \mathbb{P}^{f-1}$ is smooth irreducible non-degenerate. In particular, $F$ has only one maximal irreducible covering family of lines.
\end{enumerate}
\end{Theorem}


The following very useful result was proved in \cite[Corollary 4]{BEL}. It will play a crucial role in what follows.


\begin{Theorem}[\cite{BEL}]\label{criterion} Let $X\subset\p^N$ be as in {\rm($*$)}. If 
$$-K_X=\O_X(n+1-d),$$
then $X\subset\p^N$ is a complete intersection of type $(d_1,\ldots,d_c)$.
\end{Theorem}
\begin{proof} Under our hypothesis 
\begin{align*}h^{n+1}(\I_X(d-n-1))&=h^n(\O_X(d-n-1))\\
&=h^0(\omega_X\otimes\O_X(n+1-d))=h^0(\O_X)\neq 0.\end{align*}
Thus $X\subset\p^N$ fails to be $d$-regular and we can apply \cite[Corollary 4]{BEL}.
\end{proof}


Now we can prove the following theorem, showing that $\mathcal{L}_x\subset \p^{n-1}$ ``behaves better'' than
$X\subset \p^N$. Indeed we can control the number of equations defining $\mathcal{L}_x$, as in the classical Example~\ref{ex}~(1), but without assuming $X$ to be a complete intersection.


\begin{Theorem}\label{schemeLxrefined} Let $X\subset\p^N$ be as in {\rm($*$)}. For $x\in X$ a general point, put $a=\dim(\mathcal L_x)$. Then the following results hold: 

\begin{enumerate}

\item If $\mathcal L_x\subset\p^{n-1}$ is nonempty, it is set theoretically defined by (at most) $d$ equations; in particular, we have $a\geq n-1-d$.

\item If $X$ is quadratic and $\mathcal L_x\subset\p^{n-1}$ is nonempty, $\mathcal L_x$ is scheme theoretically defined by (at most) $c$ quadratic equations.

\item If $d\leq n-1$, then $\mathcal L_x\neq\emptyset$; assume moreover that $n\geq c+2$ if $X$ is quadratic. Then

\begin{enumerate}

\item $X\subset\p^N$ is a Fano manifold with $\Pic(X)\simeq\mathbb Z\langle H\rangle$ and $i(X)=a+2$;

\item the following conditions are equivalent:

\begin{enumerate}

\item[{\rm (i)}]  $X\subset\p^N$ is a complete intersection;
\item[{\rm (ii)}] $\mathcal L_x\subset\p^{n-1}$ is a complete intersection of codimension $d$;
\item[{\rm (iii)}] $a=n-1-d$.

\end{enumerate}
\end{enumerate}

\item Assume that $\Pic(X)\simeq\mathbb Z\langle H\rangle$, $a\geq \frac{n-1}{2}$ and $\mathcal L_x\subset\p^{n-1}$ is a complete intersection. Then $X$ is conic-connected, $a\leq n-c-1$ and $n\geq 2c+1$.


%\item
%Assume that $d\leq \frac{n-1}{2}$. Then $\mathcal L_x\subset\p^{n-1}$ is a smooth irreducible non-degenerate complete intersection and
%$$\frac{n-1}{2}\leq n-1-d\leq a\leq n-c-1.$$

\end{enumerate} 
\end{Theorem}
\begin{proof} 
Let us take a closer look at the proof of the previous result given in \cite{BEL}. Since $X\subset\p^N$ is scheme theoretically defined by equations of degree $d_1\geq d_2\geq\cdots\geq d_m$, we can choose $f_i\in H^0(\p^N,\I_X(d_i))$, $i=1,\ldots,c$,
such that, letting $Q_i=V(f_i)\subset\p^N$, we obtain the complete intersection scheme
$$Y=Q_1\cap\cdots\cap Q_c=X\cup X',$$
where $X'$ (if nonempty) meets $X$ in a divisor.

Thus for $x\in X$ general we have that  a line $l$ passing through $x$ is contained in $X\subseteq Y$ if and only if $l$ is contained in $Y$, that is $\mathcal L_x(X)\subset\p^{n-1}$ coincides set theoretically with $\mathcal L_x(Y)\subset\p^{n-1}$.
Suppose, as recalled above, that
$$Y=V(f_1,\ldots, f_c)=X\cup X'\subset\p^N$$ with $f_i$ homogeneous polynomials of degree $d_i$ and let $x\in X$ be a general point. Without loss of generality we can suppose $x=(1:0:\ldots:0)$. Then in the affine space $\mathbb A^N$ defined by $x_0\neq 0$, we have
$f_i=f_i^1+f_i^2+\cdots+f_i^{d_i}$, with $f_i^j$ homogeneous of degree $j$ in the variables $(y_1,\ldots, y_N)$, where  $y_l=\frac{x_l}{x_0}$ for every $l\geq 1$ and $x=(0,\ldots,0)$. The equations of $\p^{n-1}$ inside
 $\p^{N-1}=\p((\mathbf T_x\p^N)^*)$ are exactly $f_1^1=\cdots=f_c^1=0$ while the
equations of $\mathcal L_x(Y)$ inside $\p^{n-1}$ are  $f_1^2=\cdots=f_1^{d_1}=\cdots=f_c^2=\cdots=f_c^{d_c}=0$, which are exactly $\sum_{i=1}^c(d_i-1)=d$. In particular $\mathcal L_x\subset\p^{n-1}$,
if not empty, is set theoretically
defined by these equations, proving (1).

To prove (2), assume that $X$ is quadratic. Keeping the notation above, we see that $\mathcal L_x\subset\p^{n-1}$ is scheme theoretically defined by the equations
$f_1^2,\ldots, f_m^2$ (modulo the ideal generated by $f_1^1,\ldots, f_c^1$). But the same homogeneous quadratic equations define the affine scheme $X\cap \mathbf{T}_xX\subset \mathbf{T}_xX$. In particular, the pointed affine cone over $\mathcal L_x\subset\p^{n-1}$ and the scheme $(X\cap \mathbf{T}_xX) \setminus \{x\}$ coincide. Consider now, as above, $Y=V(f_1,\ldots, f_c)=X\cup X'$. We have that the pointed affine cone over $\mathcal L_x(Y)\subset\p^{n-1}$ and the scheme $(Y\cap \mathbf{T}_xY) \setminus \{x\}$ coincide and are scheme theoretically defined by (at most) $c$ quadratic equations. But $X$ and $Y$ coincide in a neighborhood of $x$, hence the pointed affine cones over $\mathcal L_x(X)$ and $\mathcal L_x(Y)$ also coincide. As $\mathcal L_x(X)$ is smooth, this implies that $\mathcal L_x(Y)$ is also smooth and so they coincide as schemes. This shows (2).

Under the hypothesis of (3), we deduce 
$a\geq n-1-d\geq 0.$

The assertion about the Picard group in (a) follows from the Barth--Larsen Theorem, see \cite{BL}.
Since $x\in X$ is general and since $\mathcal L_x\neq\emptyset$ by the above argument, for every line $l$ passing through $x$ we have  $$-K_X\cdot l= 2+a\geq 2,$$
concluding the proof of (a).

To show (b), let us remark that by the previous discussion (i) implies (ii); (ii) implies (iii) is obvious.  Theorem \ref{criterion} shows that (iii) implies (i).

We now pass to part (4). Since $a\geq \frac{n-1}{2}$,
$\mathcal{L}_x\subset \p^{n-1}$ is smooth irreducible and non-degenerate by Theorem \ref{reduction} (3).
As $\mathcal{L}_x\subset \p^{n-1}$ is a non-degenerate complete intersection of dimension greater or equal to one less its codimension, it follows from the Terracini Lemma that $S\mathcal{L}_x=\p^{n-1}$, see e.g. \cite{Russo}. Now we may apply \cite[Theorem~3.14]{HK} to infer that $X$ is conic-connected. In particular, we have $\delta(X)>0$. From Proposition~\ref{secondform} it follows that $\dim|II_{x,X}|=c-1$. As $\mathcal{L}_x$ is contained in the base-locus of $|II_{x,X}|$, it follows that $\mathcal{L}_x$ is contained in at least $c$ linearly independent quadrics. But, being a complete intersection, the number of such quadrics can not exceed the codimension of $\mathcal{L}_x$ in $\p^{n-1}$.
This means that $c\leq n-1-a$, which is what we wanted.\end{proof}

%To show (4), we first observe that by part (1) we have $a\geq n-1-d\geq \frac{n-1}{2}$. Next, since $d\geq c$ we get $n\geq 2c+1$, so $\Pic(X)\simeq\mathbb Z\langle H\rangle$ by \cite{BL}. From Theorem \ref{reduction}(3) $\mathcal{L}_x\subset \p^{n-1}$ is smooth irreducible and non-degenerate. Being defined scheme theoretically by $d\leq \frac{n-1}{2}$ equations, it is a complete intersection by the result of Faltings \cite{Faltings}. The last inequality follows from (3).




Next we investigate manifolds covered by lines when $a\geq n-c$. In this case we show that for each line from some
covering family $\mathcal F$ having $\dim(\mathcal{F}_x)=a$, there is some hyperplane which is tangent to $X$ along it; in particular $X$ can not be a complete intersection!

\medskip

We refer to the setting in ($*\!*\!*$). The following considerations are an elaboration of Lemma 1 from \cite{Bu}.

Consider a line $l\subset X$. Then $N_{l/X}\simeq\O_{\p^1}(a_1)\oplus\cdots\oplus\O_{\p^1}(a_{n-1})$ with $a_1\leq a_2\leq\cdots\leq a_{n-1}\leq 1$. Let $a(l)=\#\{i\mid a_i=1\}$.
If $l$ passes through a general point $x\in X$, then $a_1\geq 0$ and $\deg(N_{l/X})=a(l)$.

For every line $l\subset X$ we shall consider the following linear spaces:
\begin{equation}
\big\langle {\textstyle\bigcup\limits_{x\in l}}T_xX\big\rangle,
\end{equation}
which is the linear span of the union of the tangent spaces at points $x\in l$
and 
\begin{equation}
{\textstyle\bigcap\limits_{x\in l}}T_xX.
\end{equation}

By the discussion from ($*\!*\!*$), we also have the equality
\begin{equation}\label{Tint}
\pi_1(\p((\mathcal P_X)_{|l}))= {\textstyle\bigcup\limits_{x\in l}}T_xX.
\end{equation}

Since we shall be interested in $\bigcup_{x\in l}T_xX$ and in the dimension of its linear span in $\p^N$, we shall analyze  $\mathcal P_{X|l}$,
where for the moment $l\subset X$ is an arbitrary line. From \eqref{princ} we deduce that $\mathcal P_X$  and $\mathcal P_{X|l}$ are generated by global
sections so that
\begin{equation}\label{splittingarbitrary}
\mathcal P_{X|l}\simeq \bigoplus\limits_{j=1}^{n+1-b_0(l)}\O_{\p^1}(b_j(l))\bigoplus \O_{\p^1}^{b_0(l)}
\end{equation}
with $b_j(l)>0$ for every $j=1,\ldots, n+1-b_0(l)$. 

Let $\Pi=\p^b=\p(U)\subset\p^N$. We have a surjection $V\to U$ inducing a surjection $V\otimes\O_X\to U\otimes\O_X$ and hence
an inclusion, over $X$,  $\p(U)\times X\subset\p(V)\times X$. Given a subvariety $Y\subseteq X$ we have that $\Pi\subset T_yX$ for every
$y\in Y$ if and only if the natural surjection $V\otimes\O_Y\to U\otimes \O_Y$ factorizes through $V\otimes \O_Y\to \mathcal P_{X|Y}$, that is if and only if there exists  a surjection $\mathcal P_{X|Y}\to U\otimes \O_Y$.
Thus for a line $l\subset X$ we obtain that if $\Pi=\p^b\subset T_xX$ for every $x\in l$, then $b\leq b_0(l)-1$.




\begin{Proposition}{\rm(cf. \cite[Lemma1]{Bu})}\label{contact} Let notation be as above and let $l\subset X$ be a line passing through a general point of $X$. Then:
\begin{enumerate}
\item 
$$\mathcal P_{X|l}\simeq\O_{\p^1}^{a(l)+2}\oplus\O_{\p^1}(1)^{n-1-a(l)}$$
and 
$$\dim\big(\big\langle {\textstyle\bigcup\limits_{x\in l}}T_xX\big\rangle\big)=N-\ho(N_{X/\p^N}^*(1)_{|l}).$$

In particular the variety $\bigcup_{x\in l}T_xX$ is isomorphic to a linear projection
of a cone with  vertex a linear space of dimension $a(l)+1$ over the Segre variety $\p^1\times \p^{n-2-a(l)}$
so that $$\dim\big(\big\langle {\textstyle\bigcup\limits_{x\in l}}T_xX\big\rangle\big)\leq\min\{N, 2n-1-a(l)\}.$$


\item $$\dim\big({\textstyle \bigcap\limits_{x\in l}}T_xX\big)=a(l)+1$$ and for $x,y\in l$ general we have
$$T_y\widetilde C_x={\textstyle\bigcap\limits_{z\in l}}T_zX,$$ where $\widetilde C_x$ is the irreducible component of the locus
of lines through $x$ to which $l$ belongs.



\item $\ho(N_{X/\p^N}(-1)_{|l})=N-a(l)-1$;


\item If $N\geq 2n-1-a(l)$, then $\ho(N_{X/\p^N}^*(1)_{|l})\geq N+a(l)-2n+1$ with equality holding if and only if 
$\dim(\langle \bigcup_{x\in l}T_xX\rangle)= 2n-1-a(l)$.
\end{enumerate}
\end{Proposition}
\begin{proof}  From the exact sequence
$$0\to \O_l(-1)\to \mathcal P_{X|l}^*\to T_X(-1)_{|l}\to 0$$
and from $h^0(T_X(-1)_{|l})=a(l)+2$ we deduce $b_0(l)=h^0(\mathcal P_{X|l}^*)=a(l)+2$. Moreover
\begin{align*}n+1-b_0(l)\leq\sum_{j=1}^{n+1-b_0(l)}b_j(l)&=\deg(\mathcal P_{X|l})=(K_X+(n+1)H)\cdot l\\
&=-a(l)-2+n+1=n+1-b_0(l)\end{align*}
so that $b_j(l)=1$ for every $j=1,\ldots,n-1-a(l)$, proving the first assertion.


Let $\Pi=\bigcap_{x\in l}T_xX=\p^b$.  By the previous analysis we deduce $b\leq a(l)+1$. If $l$ passes through the general point $x\in X$,
consider $\widetilde C_x$, the irreducible component of the locus of lines through $x$ to which $l$ belongs, which is a cone
whose vertex contains $x$. We know that $\dim(\widetilde C_x)=a(l)+1$ and that for every $y\in l$ obviously $T_y\widetilde C_x\subset T_yX$.
Since $T_y\widetilde C_x$ does not depend on $y\in l$ for $y\in l$ general, we deduce that, for $x,y\in l$ general,
$T_y\widetilde C_x\subseteq\bigcap_{z\in l}T_zX.$
Therefore $a(l)+1\leq b$ which together with the previous inequality yields $b=a(l)+1$ and
\begin{equation}\label{intformula}
T_y\widetilde C_x={\textstyle \bigcap\limits_{z\in l}}T_zX
\end{equation}
for $x,y\in l$ general points. 

 The other implications easily follow from the exact sequences in ($*\!*\!*$) and their geometrical interpretation.  
\end{proof}


%\begin{Remark} The above proof provides a description of the projective tangent space to the Hilbert scheme of lines
%$C_x\subset\p((\mathbf T_xX)^*)=\p^{n-1}$ at the point $[l]\in C_x$ as the projectivization of $\mathbf T_x\widetilde C_y$, $y\in l$ a general point.
%Moreover $T_x\widetilde C_y$ is the projectivization of the non-negative part of $\mathcal P_{X|l}^*$.
%\end{Remark}



\section{Defective manifolds}

We begin by recalling the main known results about LQEL manifolds; complete proofs of the statements in the next theorem may be found in \cite{Ru, IR, IR2, Fu}.
\begin{Theorem}\label{LQEL} Assume $X\subset \mathbb{P}^N$ is a LQEL manifold of secant defect $\delta$. Then:

\begin{enumerate}
\item $X$ is a Fano rational manifold with ${\rm rk Pic}(X)\leq 2$.



\item If ${\rm rk Pic}(X)=2$ then $X$ is one of: 



\begin{enumerate}
\item $\mathbb{P}^a\times \mathbb{P}^b$ in its Segre embedding, or
\item the hyperplane section of the above, or
\item the blowing-up of $\p^n$ with center a linear space $L$, embedded by the linear system of quadrics through $L$.
\end{enumerate}


\item If ${\rm rk Pic}(X)\!= \!1$, $X\!\cong\! v_2(\mathbb{P}^n)$ or ${\rm Pic}(X)\! = \!\mathbb{Z} \langle H\rangle$ and $i(X) = \frac {n+\delta}{2}$.




\item If $\delta \geq 3$ then $\mathcal{L}_x \subset \mathbb{P}^{n-1}$ is again a LQELM, $S\mathcal{L}_x= \mathbb{P}^{n-1}$, $\dim (\mathcal{L}_x)= \frac{n+\delta}2 -2$, $\delta (\mathcal{L}_x) =\delta -2$. If $\delta \geq \frac n2$, a complete classification is obtained. 



\item $X$ is a complete intersection if and only if $X$ is a quadric $\mathbb{Q}^n (\delta=n)$.



\item If $X\not \cong \mathbb{Q}^n$ then $\delta \leq \frac{n+8} 3$; equality cases are classified.
\end{enumerate}
\end{Theorem}



\begin{Conjecture}{\rm(\cite{IR2})} Assume $X$ is a LQELM with ${\rm Pic}(X) \cong \mathbb{Z}\langle H\rangle$. Then $X$ is obtained by linear sections and/or isomorphic projections from a rational homogeneous manifold, in its natural minimal embedding.\end{Conjecture}

We recall that rational homogeneous manifolds are well understood. In particular, those which are secant defective are known to be LQEL manifolds and are completely classified, see \cite{Ka, Zak}. They turn out to be quadratic manifolds and moreover we have that $\delta \leq 8$ if $X$ is not a quadric. So we expect any linearly normal LQEL manifold to be quadratic. Conversely, any quadratic secant defective manifold whose Koszul syzygies are generated by linear ones is a LQEL manifold~by~\cite{Ve}.

\medskip

Next we consider the case of dual defective manifolds. Working in different settings, Mumford and Landman called the attention on these very special but intriguing class of embedded manifolds. They have since then been studied thoroughly by Ein in \cite{Ein, Ein2} and by Beltrametti--Fania--Sommese in \cite{BFS}. See also \cite{LS, Munoz, Munoz2}. 

For $X\subset\p^N$, let $X^*\subset\p^{N*}$ be the dual variety of $X$ and let $\defect(X)=N-1-\dim(X^*)$ be the {\it dual defect} of $X$.

For a hyperplane $H\subset\p^N$  we define the {\it contact locus} of $H$ on $X$ as
\begin{equation}\label{contactlocus}
L=L(H)=\{x\in X\mid T_xX\subseteq H\}=\Sing(X\cap H)\subset X.
\end{equation}
If $[H]\in X^*$  corresponds to a smooth point in $X^*$, then by Reflexivity $L(H)\simeq \p^{\defect(X)}$ is an embedded linear subspace of $\p^N$ contained in $X$. Any line $l\subset L(H)$ with $[H]\in X^*$ 
is called a {\it contact line} on $X$. We recall a basic result on the geometry of contact lines proved by Ein in \cite[\S 2]{Ein}.

\begin{Theorem}[\cite{Ein}] \label{Eindual} 
Let $k=\defect(X)>0$ and let $l\subset L(H)$ be a contact line with $[H]\in X^*$ general. Then
$N_{l/X}\simeq\O_{\p^1}^{\frac{n-k}{2}}\oplus\O_{\p^1}(1)^{\frac{n+k-2}{2}};$
in particular, $n$ and $k$ have the same parity, as first proved by Landman.

\end{Theorem}



Let $k=\defect(X)>0$ and let $x\in X$ be a general point. By Theorem~\ref{Eindual} there exists a unique irreducible component of $\mathcal L_x$ of dimension $\frac{n+k-2}{2}$ containing a given general contact line.
The union of all these irreducible components of $\mathcal L_x$ is called $C_x$. Since $\mathcal L_x$ is smooth and since $\frac{n+k-2}{2}\geq \frac{n-1}{2}$, there exists a unique
irreducible component of $\mathcal L_x$ containing all the general contact lines passing through $x$, i.e. $C_x\subseteq \mathcal L_x$ is an irreducible component of $\mathcal L_x$ of dimension
$\frac{n+k-2}{2}$ and
hence an irreducible smooth subvariety of $\mathcal L_x$. Easy examples like the Segre varieties $\p^1\times\p^{n-1}\subset\p^{2n-1}$ show that in general $C_x\subsetneq \mathcal L_x$.
For every line $[l]\in C_x$, we get $a(l)=\frac{n+k-2}{2}$.
 



\begin{Proposition}\label{contactline} Let $X\subset\p^N$ be as in {\rm($*$)} with  $\defect(X)=k>0$ and let $x\in X$ be a
general point. If $k\geq n-2c+2$, then for a general  line $[l]\in C_x$
 \[\dim\big(\big\langle {\textstyle \bigcup\limits_{x\in l}}T_xX\rangle)=\frac{3n-k}{2}\quad
\mbox{and}\quad
N_{X/\p^N}(-1)_{|l}\simeq\O_{\p^1}^{\frac{k+2c-n}{2}}\oplus\O_{\p^1}(1)^{\frac{n-k}{2}}.\]
Moreover,  every  line $[l]\in C_x$  is a contact line and for a general $[l]\in C_x$  every hyperplane $H$ such that
$l\subset L(H)$ represents a smooth point of $X^*$. In particular  through a general line $[l]\in C_x\subset\p^{n-1}$ there passes a linearly embedded $\p^{k-1}$
contained in $C_x$, corresponding to lines in $L(H)$ passing through $x$. 

\end{Proposition}
\begin{proof} If $[l]\in C_x$, then $a(l)=\frac{n+k-2}{2}$
so that the hypothesis and  Proposition \ref{contact}
imply 
\begin{equation}\label{dimcont}
\dim\big(\big\langle {\textstyle \bigcup\limits_{x\in l}}T_xX\rangle)\leq 2n-1-a(l)\leq  N-1.
\end{equation}
Thus every  line $[l]\in C_x$  is contained in the contact locus of at least one hyperplane $H$ with $[H]\in (T_xX)^*$.

Let $X^*_x=\{[H]\in X^*\mid T_xX\subseteq H\}=(T_xX)^*\simeq\p^{c-1}$.  
Put
$$Z_x=\{([l],[H])\mid  l\subset L(H)\}\subset C_x\times X^*_x,$$
where $p_x:Z_x\to C_x$ is the restriction of the first projection and $q_x:Z_x\to X^*_x$ the restriction
of the second projection.  For every $[H]\in \Sm(X^*)\cap X^*_x$ we have  $q_x^{-1}([H])=\p^{k-1}$, where $\Sm(X^*)$ denotes the smooth locus of $X^*$. Thus 
$q_x$ is surjective and every irreducible component of $Z_x$ dominating $X^*_x$ has dimension $k+c-2$.
By \eqref{dimcont} also $p_x$ is surjective and by definition of $Z_x$ and $C_x$ every irreducible component
of $Z_x$ dominating $C_x$ dominates $X^*_x$ and vice versa. Let $U=q_x^{-1}( \Sm(X^*)\cap X^*_x)$. Since $p_x$ is proper
and surjective, there exists an open subset $V\subseteq C_x$ such that $p_x^{-1}(V)\subseteq U$. Thus for every $[l]\in V$,
every hyperplane $H$ such that $l\subset L(H)$ is a smooth point of $X^*$ and through $[l]$ there passes a linear embedded
$\p^{k-1}\subset\p^{n-1}$ contained in $C_x$.


Let $[l]\in C_x$ be an arbitrary line and let $F_{[l]}=p_x^{-1}([l])$. By definition 
$F_{[l]}=(\langle \bigcup_{x\in l}T_xX\rangle)^*=\p^{N-b(l)-1}$, where $b(l)=\dim(\langle \bigcup_{x\in l}T_xX\rangle)$.
Thus for a general $[l]\in C_x$
\begin{equation}\label{dimF}
\dim(F_{[l]})=k+c-2-\frac{n+k-2}{2}=\frac{k+2c-2-n}{2}, 
\end{equation}
and
\begin{equation}
n+c-b(l)-1=N-b(l)-1=\dim(F_{[l]})=\frac{k+2c-2-n}{2},
\end{equation}
yielding $b(l)=\dim (\langle \bigcup_{x\in l}T_xX\rangle) =\frac{3n-k}{2}=2n-1-a(l)$.

The locally free sheaf $N_{X/\p^N}(-1)_{|l}$ has rank $c$ and is generated by global sections, so that
$$N_{X/\p^N}(-1)_{|l}\simeq\bigoplus_{j=1}^{c}O_{\p^1}(d_j),$$
with $d_j\geq 0$. Let $d_0(l)=\#\{i\mid d_i=0\}$ and let $d_1\leq\cdots \leq d_c$. 
The exact sequences 
$$0\to N_{L/X}(-1)_{|l}\to \O_l^{N-k}\to N_{X/\p^N}(-1)_{|l}\to 0$$ and
$$0\to \O_l^{k-1}\to N_{l/X}(-1)\to N_{L/X}(-1)_{|l}\to0$$
together with Theorem~\ref{Eindual} yield 
$$\frac{n-k}{2}=-\deg(N_{l/X}(-1))=\deg(N_{X/\p^N}(-1)_{|l})=\sum_{j=d_0(l)+1}^c d_j\geq c-d_0(l).$$
Thus $d_0(l)\geq c-\frac{n-k}{2}=\frac{k+2c-n}{2}$ with equality holding if and only if $d_r=1$ for every $r=d_0(l)+1,\ldots ,c$.

By definition of $d_0(l)$ and by Proposition \ref{contact} part (4) we get
\begin{equation}\label{stima1}
d_0(l)=\ho(N_{X/\p^N}^*(1)_{|l})=N+a(l)-2n+1=\frac{k+2c-n}{2},
\end{equation}
\samepage{so that 
$N_{X/\p^N}(-1)_{|l}\simeq\O_{\p^1}^{\frac{k+2c-n}{2}}\oplus\O_{\p^1}(1)^{\frac{n-k}{2}},$
as claimed.}
\end{proof}


\begin{Lemma}\label{3k}
Let $X\subset\p^N$ be as in {\rm($*$)} with  $\defect(X)=k>0$. If $k\geq\frac{n}{3}$,
then $k\geq n-2c+2$.
\end{Lemma}
\begin{proof} By Zak's Theorem on Tangencies, 
$k\leq c-1$. Then
$$n\leq 3k\leq k+2c-2,$$
concluding the proof.
\end{proof}


Let us recall that $X\subset \p^N$ is an {\it $r$-scroll}, if $X\simeq \p(E)$ where $E$ is a rank~$r+1$ vector bundle over some manifold $W$ and the fibers of the projection $\pi:\p(E)\to W$ are linearly embedded in $\p^N$. When $r > \dim(W)$, the scroll $X$ is dual defective and its defect equals $r-\dim(W)$. The next result characterizes scrolls among all dual defective manifolds. 
%\ro[In the next section we shall relate this result
%to various versions of Hartshorne Conjecture on Complete Intersections.]

\begin{Theorem}\label{scrollimit}
Let $X\subset\p^N$ be as in {\rm($*$)} with  $\defect(X)=k>0$. Let $C_x\subset\p^{n-1}$ be as above and put
$\p^m\simeq T \subseteq\p^{n-1}$ to be the linear span of $C_x$ in $\p^{n-1}$. The following conditions are equivalent:
\begin{enumerate}
\item[{\rm (i)}] $X$ is an $\frac{n+k}{2}$-scroll over a manifold of dimension $\frac{n-k}{2}$;
\item [{\rm (ii)}] $\dim(C_x)>2\codim_T(C_x)$ (or equivalently $k>\frac{4m+6-3n}{3}$).
\end{enumerate}
\end{Theorem}
\begin{proof} We only have to prove that (ii) implies (i). Since $n\equiv k$ modulo 2, it follows that $\dim(C_x)=\frac{n+k-2}{2}\geq[\frac{n}{2}]$. In particular the lines in $C_x$ generate
an extremal ray of $X$ by Theorem~\ref{reduction}~(1). Let $\phi:X\to W$ be the contraction of this ray and let $F$ be a general fiber
of $\phi$. Then $F\subset\p^N$ is a smooth irreducible projective variety such that $\Pic(F)\simeq\mathbb Z\langle H_F\rangle$. Let $f=\dim(F)$ and let $\langle F\rangle$ be the linear span of $F$ in $\p^N$.
Then $C_x\subseteq \p((\mathbf T_xF)^*)=\p^{f-1}$ is smooth irreducible non-degenerate by Theorem~\ref{reduction}~(3).
Thus $m=f-1$. Moreover, by \cite[Theorem (1.2)]{BFS} we have $\defect(F)=k(F)=k+n-f$ so that the hypothesis in (ii) yields $k(F)>\frac{f+2}{3}$.
By Lemma \ref{3k} and Proposition \ref{contactline} every line in $C_x$ is a contact line for $F\subseteq \langle F\rangle$ and $C_x\subseteq\p^{f-1}$
is covered by linear spaces of dimension $k(F)-1>[\frac{\dim(C_x)}{2}]$. From \cite{Sato} (see also \cite{BI} for a simple proof in the spirit of the present paper) it follows that $C_x\subseteq \p((\mathbf T_xF)^*)=\p^{f-1}$ is
a scroll. The condition in (ii) and the Barth-Larsen Theorem, see \cite{BL}, give $\Pic(C_x)\simeq\mathbb Z\langle H_{C_x}\rangle$. So we get $C_x=\p^{f-1}$, hence $F=\p^f$ and $k(F)=f$. Therefore $f=\frac{n+k}{2}>\frac{n}{2}$ and  $X\subset\p^N$ is a scroll
by \cite[Theorem 1.7]{Ein2}.
\end{proof}



\begin{Corollary}\label{extremal} Let $X\subset\p^N$ be as in {\rm($*$)}, with  $\defect(X)=k>0$. Let $\phi:X\to W$ be the contraction whose existence is ensured by Theorem~{\rm\ref{reduction}~(1)}.
Assume that $X$ is not
a scroll. Then $$k\leq \frac{n+2-4\dim(W)}{3}\leq \frac{n+2}{3}.$$
Moreover:
\begin{enumerate}
\item $k=\frac{n+2}{3}$ if and only if\, $N=15$, $n=10$ and $X\subset\p^{15}$
is projectively equivalent to the $10$-dimensional spinorial variety $S^{10}\subset\p^{15}$.
\item If $\dim(W)>0$,  
then  $k=\frac{n+2-4\dim(W)}{3}$ if and only
if\, $\dim(W)=n-10\leq 3$ and $\phi:X\to W$ is a fibration such that the  
general fiber $F\subset\langle F\rangle \subset\p^N$ is isomorphic to  
$S^{10}\subset\p^{15}\subset\p^N$.

\end{enumerate}

\end{Corollary}
\begin{proof} Keeping the notation from the preceding theorem, we have  $k\leq\frac{4m+6-3n}{3}$, $m=f-1$ and $f=n-\dim(W)$. So we may assume that $k=\frac{n+2}{3}$. 
Then $\dim(C_x)=\frac{2(n-1)}{3}$. By Lemma \ref{3k} and
Proposition \ref{contactline} $C_x\subset\p^{n-1}$ is covered by linear spaces of dimension $k-1=\frac{n-1}{3}=\frac{\dim(C_x)}{2}$. Then by \cite{Sato} and \cite{ON}
the variety $C_x\subset\p^{n-1}$ is one of the following:
\begin{enumerate}
\item[{\rm a)}] a scroll;
\item[{\rm b)}] a quadric hypersurface of even dimension;
\item[{\rm c)}] $\mathbb G(1,r)$ Pl\" ucker embedded with $r\geq 4$, or one of its isomorphic projections.
\end{enumerate}

If $n<7$, then $n=4$ and $k=n-2$ so that $X\subset\p^n$ would be a scroll over a curve by \cite[Theorem 3.2]{Ein}, contradicting our
assumption. If $C_x\subset\p^{n-1}$ is a (quadric) hypersurface, then $\frac{2(n-1)}{3}=n-2$ yields
$n=4$, obtaining once again a contradiction. Thus we can suppose $n\geq 7$, so that $\Pic(C_x)=\mathbb Z\langle H\rangle$ by the Barth-Larsen Theorem. Therefore
we are necessarily  in case c). Since the secant defect of $\mathbb G(1,r)$ is four, we get:
$n-1\geq 2\dim(C_x)+1-\delta(C_x)=\frac{4n-13}{3}$, yielding $n\leq 10$. On the other hand, we have $\dim(C_x)=2(r-1)\geq 6$, so that $n\geq 10$. Therefore $n=10$, $k=4$ and $i(X)=8$; moreover, the proof of the previous theorem shows that $F=X$, i.e. $\Pic(X)\simeq\mathbb Z\langle H\rangle$. The conclusion of part (1) follows by Mukai's
classification of prime Fano manifolds of index $n-2$, see \cite{Mukai}.

To prove (2), assume that $k=\frac{n+2-4\dim(W)}{3}$. By \cite[Theorem (1.2)]{BFS}
$$k(F)=\frac{n+2-4\dim(W)}{3}+\dim(W)=\frac{f+2}{3},$$
so that by the first part $f=10=n-\dim(W)$ and $F\subset\langle F 
\rangle \subset\p^N$ is isomorphic to $S^{10}\subset\p^{15}$.
\end{proof}

\begin{Remark} The preceding corollary improves one of the main results in \cite{Ein2} stating that if $k>0$ and $X$ is not a scroll, we have $k\leq \frac{n-2}{2}$, with equality only if $X$ is projectively isomorphic either to $\mathbb G(1,4)\subset \p^9$ or to $S^{10}\subset \p^{15}$. The next corollary is the main result of \cite{Ein}, proved in a different manner. Another proof, based on the theory of LQEL manifolds, may be found in \cite{IR2}. 
\end{Remark}




\begin{Corollary}\label{Einex}{\rm (\cite[Theorem 4.5]{Ein})}  Let $X\subset\p^N$ be a smooth irreducible non-degenerate variety such  that $\dim(X)=\dim(X^*)$.
If $n\leq 2c$, then $X$ is projectively
equivalent to one of the following:
\begin{enumerate}
\item[(a)] a smooth  hypersurface $X\subset\p^{n+1}$, $n=1, 2$;

\item[(b)] a Segre variety $\p^1\times\p^{n-1}\subset\p^{2n-1}$;

\item[(c)] the Pl\" ucker embedding $\mathbb{G}(1,4)\subset\p^9$;

\item[(d)] the $10$-dimensional spinorial variety
$S^{10}\subset\p^{15}$.
\end{enumerate}
\end{Corollary}
\begin{proof} By hypothesis $k=c-1$.
Clearly $k=0$ if and only if
$X\subset\p^{n+1}$ is a hypersurface, giving case (a).
 From now on we suppose
$k>0$. By \cite[Theorem~3.2]{Ein} $k=n-2$ and $N=2n-1$ if and only if we are in
case  (b); see also \cite[Theorem~3.3,~c)]{Ein}. If $k\leq n-3$, then $n\geq c+2$ so that by the Barth--Larsen Theorem we deduce $\Pic(X)\simeq\mathbb Z\langle H\rangle$.
Hence we can assume that $X\subset\p^N$ is not a scroll. Then by Corollary \ref{extremal} either we are in case (d) or we have $\frac{n+2}{3}> k=c-1\geq \frac{n-2}{2}$. Taking into account that we have $n\equiv k$ modulo $2$, this leads to only one more case, namely $n=6, k=2$. So we also have $i(X)=5, c=3, N=9$ and we are in case (c) by Fujita's classification of del Pezzo manifolds, see \cite{Fujita}.
\end{proof}

\begin{Conjecture}\label{DD}{\rm(\cite{IR2})} Any dual defective manifold with cyclic Picard group is a LQELM.
\end{Conjecture}
 
 Theorem~(1.2) from \cite{BFS}, based on Theorem~\ref{reduction}~(1), reduces the study of dual defective manifolds to the case when $\Pic(X)\simeq\mathbb Z\langle H\rangle$. Note that $k\leq 4$ in all known examples other than scrolls.


\begin{Proposition} Conjecture~\ref{DD} for a dual defective manifold $X\subset \p^N$ as in {\rm($*$)} with dual defect $k$ is equivalent to the conjunction of the following two assertions:
\begin{enumerate}
\item $X$ is conic-connected, 
and
\item $k\geq n-c-1$.
\end{enumerate}
\end{Proposition}
\begin{proof} If $X$ is a LQELM, of defect $\delta$, $\Pic(X)\simeq\mathbb Z\langle H\rangle$ and $X$ is dual defective of defect $k$, it follows that $\delta=k+2$.
Indeed, we have $$\frac{n+\delta-4}{2}=\dim(\mathcal{L}_x)=\frac{n+k-2}{2}.$$
Assume now that $X$ is dual defective and (1) and (2) hold. Since $X$ is conic-connected, it follows from \cite[Proposition~3.2]{IR2} that $X$ is a LQELM if $i(X)\geq \frac{n+\delta}{2}$. Replacing $X$ by one of its isomorphic projections allows us to assume that $SX=\p^N$, hence $\delta=n-c+1$. Therefore the inequality in (2) becomes $k+2\geq \delta$. But we have
$$i(X)=\dim(\mathcal{L}_x)+2=\frac{n+k+2}{2}\geq \frac{n+\delta}{2},$$
so that $X$ is a LQELM.

Conversely, if $\Pic(X)\simeq\mathbb Z\langle H\rangle$ and $X$ is (dual defective and) a LQELM, we trivially have condition (1). As for (2), by the above we have $k=\delta-2$; so condition (2) becomes $\delta\geq n-c+1$, which is obvious.
\end{proof}
\begin{Remarks} 
\begin{enumerate}
\item Condition (2) above implies the inequality $k\geq n-2c+2$, which is the hypothesis in Proposition~\ref{contactline} (use that $n\equiv k$ modulo $2$).
\item By Zak's Theorem on Tangencies $k\leq c-1$, which combined with the inequality in (2) yields $n\leq 2c$ for dual defective manifolds. Note that this was a hypothesis in Corollary~\ref{Einex}.
\item Knowing that (2) holds would lead to a much simpler proof of Corollary~\ref{extremal}, without making use of the elaborate results from \cite{Sato, ON}.

\item If $X$ is both dual defective and a LQELM, we have seen that $\delta=k+2$; therefore, the upper bounds on $\delta$ in Theorem~\ref{LQEL}, and on $k$ in Corollary~\ref{extremal}, are the same. They express the condition that $$\dim(\mathcal{L}_x)\leq 2\codim(\mathcal{L}_x, \p^{n-1}).$$ Moreover, quadrics are the only complete intersections that are LQELMs, while linear spaces (generating scrolls) are the only complete intersections among dual defective manifolds. This naturally leads us to the next section.
\end{enumerate}
\end{Remarks}

\section{Around the Hartshorne Conjecture; the quadratic case}

In \cite{Ha}, Hartshorne  made his now famous conjecture:
\begin{Conjecture}[HC] If $X\subset \p^N$ is as in {\rm($*$)} and  $n\geq2c+1$ then $X$ is a complete intersection.
\end{Conjecture}
This is still widely open, even for $c=2$.
We would like to state explicitly several weaker versions.
\begin{Conjecture}[HCF] Assume that $n\geq 2c+1$ and $X$ is Fano; then $X$ is a complete intersection.
\end{Conjecture}

\begin{Conjecture}[HCL] Assume that $X\subset\p^N$ is covered by lines with  $\dim(\mathcal L_x)\geq\frac{n-1}{2}$ and let $T=\langle \mathcal L_x\rangle\subseteq\p^{n-1}$. If $\dim(\mathcal L_x)>2\codim_T(\mathcal L_x)$, then
$\mathcal L_x\subset\p^{n-1}$ is a complete intersection. 
\end{Conjecture}


\begin{Remarks}
\begin{enumerate}
\item When $n\geq 2c+1$, by the Barth--Larsen Theorem ${\rm Pic}(X) \cong \mathbb{Z}\langle H\rangle$. In particular $K_X=bH$ for some integer $b$. So, saying that $X$ is Fano means exactly that $b < 0$; this happens, for instance, if $X$ is covered by lines.
\item The (HCF) holds when $c=2$, see \cite{BC}.
\item By Theorem \ref{reduction}, the  (HCL) concerns Fano manifolds $X\subset\p^N$ with $\Pic(X)\simeq\mathbb Z\langle H\rangle$ and of index $i(X)\geq\frac{n+3}{2}$. Dual defective  and LQEL manifolds satisfy the (HCL), by Theorem \ref{scrollimit} and Theorem \ref{LQEL}.
\item Manifolds of (very) small degree are known to be complete intersections, cf. \cite{Barth}. The (HCF) would yield the following optimal result, see \cite{Io}.

\end{enumerate}
\end{Remarks}
\begin{Conjecture}
If $n\geq degree(X)+1$, then $X$ is a complete intersection, unless it is projectively equivalent to $\mathbb G(1,4)\subset\p^9$.
\end{Conjecture}
Note that the Segre embedding $\p^1\times \p^{n-1}\subset \p^{2n-1}$ has degree $n$ and is not a complete intersection if $n\geq 3$.
 
 Prime Fano manifolds of high index tend to be complete intersections. In the next proposition we propose such bounds on the index and show how they would follow from the (HC). However, we expect this kind of result to be easier to prove than the general (HC). %\ro[in case $X$ is a del Pezzo manifold ($i(X)=n-1$), see \cite{Fujita}, or a Mukai manifold ($i(X)=n-2$), see \cite{Mukai}, we see, a posteriori,  that (HCL) holds.]
\begin{Proposition}\label{index} 
\begin{enumerate}
\item

Let $X\subset\p^N$ be a Fano manifold with $\Pic(X)\simeq\mathbb Z\langle H\rangle$ and of index $i(X)\geq\frac{2n+5}{3}$. If the {\rm(HCL)} and the {\rm(HCF)} are true,
then $X$ is a complete intersection.
\item The same conclusion holds assuming only the {\rm(HCF)}, but asking instead that $i(X)\geq \frac{3(n+1)}{4}$.
\end{enumerate}
\end{Proposition}
\begin{proof} (1) Since $i(X)>\frac{n+1}{2}$ we get that $X$ is covered by lines by Example~\ref{ex}~(2). We have $\dim(\mathcal{L}_x)=i(X)-2\geq \frac{2n-1}{3}\geq \frac{n-1}{2}$, hence $\mathcal{L}_x\subset \p^{n-1}$ is smooth irreducible non-degenerate by Theorem~\ref{reduction}~(3). Moreover, we have $\dim(\mathcal{L}_x)\geq 2\codim(\mathcal{L}_x, \p^{n-1})+1$. 
Thus $\mathcal{L}_x\subset \p^{n-1}$ is a complete intersection by the (HCL). Next we may apply Theorem~\ref{schemeLxrefined}~(4) to infer that $n\geq 2c+1$ and then use the (HCF).

(2) Since $i(X)> \frac{2n}{3}$ we have that $X$ is conic-connected by \cite{HK}. From \cite[Proposition~3.2]{IR2} we infer 
$$\frac{3(n+1)}{4}\leq i(X)\leq\frac{n+\delta}{2}.$$
This yields  $\delta>\frac{n}{2}$ so that $SX=\p^N$ by Zak's Linear Normality Theorem. Therefore $\delta=n-c+1$
and
$$\frac{3(n+1)}{4}\leq i(X)\leq\frac{n+(n-c+1)}{2}$$ 
implies
$n\geq 2c+1$, so the (HCF) applies.
\end{proof}

Note that for complete intersections $X\subset \p^N$, $\mathcal{L}_x\subset \p^{n-1}$ is also a complete intersection. We believe that for manifolds covered by lines the converse should also hold.

\begin{Conjecture}
If $X\subset \p^N$ is covered by lines and $\mathcal{L}_x\subset \p^{n-1}$ is a (say smooth irreducible non-degenerate) complete intersection, then $X$ is a complete intersection too.
\end{Conjecture}
Theorem~\ref{schemeLxrefined}~(4) shows that the above conjecture follows from the (HCF), at least when $\dim(\mathcal{L}_x)\geq \frac{n-1}{2}$.

\medskip


Mumford in his seminal series of lectures \cite{Mum} called the attention to the fact that many interesting embedded manifolds are scheme theoretically defined by quadratic equations. 



As a special case of the results in \cite{BEL}, if $X$ is quadratic, we have: 
\begin{enumerate}
\item If $n\geq c-1$ then $X$ is projectively normal;
\item If $n\geq c$ then $X$ is projectively Cohen--Macaulay.
\end{enumerate}
Our main results in this section are:
\begin{Theorem}\label{HaCo} Assume that $X\subset \p^N$ is as in {\rm($*$)} and $X$ is quadratic.
\begin{enumerate}
\item If $n\geq c+1$ then $X$ is covered by lines. Moreover, $\mathcal{L}_x\subset \p^{n-1}$
is scheme theoretically defined by $c$ independent quadratic equations.
\smallskip
\item If $n\geq c+2$ then $X$ is a Fano manifold with $\Pic(X)\simeq\mathbb Z\langle H\rangle$.
Moreover, the following conditions are equivalent:
\begin{enumerate}
\smallskip
\item[{\rm(i)}] $X\subset \p^N$ is a complete intersection;
\item[{\rm(ii)}] $\mathcal{L}_x\subset \p^{n-1}$ is a complete intersection;
\item[{\rm(iii)}] $\dim(\mathcal{L}_x)=n-1-c$;
\item[{\rm(iv)}] $N_{X/\p^N}(-1)$ is ample.
\end{enumerate}

\smallskip
\item{\rm(HC)} If $n\geq2c+1$ then $X$ is a complete intersection.
\smallskip
\item If $\Pic(X)\simeq\mathbb Z\langle H\rangle$ and $X$ is Fano of index $i(X)\geq\frac{2n+5}{3}$, then $X$ is a complete intersection.
\end{enumerate}

\end{Theorem}
\begin{proof}
We use the notation in ($*$). Moreover, denote by $a:=\dim(\mathcal{L}_x$). Since $X$ is quadratic, we have $d=c$. 

 From Theorem~\ref{schemeLxrefined}~(2) and (3) we deduce the first part in (1) and (2). It also follows that $\mathcal{L}_x\subset \p^{n-1}$ is defined scheme theoretically by at most $c$ quadratic equations. Since $n\geq c+1$, we must have $\delta\geq n-c+1 >0$. From Proposition~\ref{secondform} we infer that the quadratic equations defining  $\mathcal{L}_x$ are independent, proving (1). This also shows that, when $\mathcal{L}_x$ is a complete intersection, it has codimension $c$ in $\p^{n-1}$. From Theorem~\ref{schemeLxrefined}~(3) the equivalence between conditions (i), (ii) and (iii) is now clear. 

Assume that $N_{X/\p^N}(-1)$ is ample. We claim that $a\leq n-1-c$. Indeed, if $a\geq n-c$, from Proposition~\ref{contact} we deduce that  $\ho(N_{X/\p^N}^*(1)_{|l})>0$. Therefore the restriction $N_{X/\p^N}(-1)|_l$ can not be ample and the claim is proved. Combined with the first part of Theorem~\ref{schemeLxrefined}, this shows that $a=n-1-c$, so that $X$ is a complete intersection. The equivalence of the four conditions in (2) is thus proved.

To prove part (3), assume that $n\geq 2c+1$. We first observe that by Theorem~\ref{schemeLxrefined}~(1) we have $a\geq n-1-c\geq \frac{n-1}{2}$. Next, since $n\geq 2c+1$, $\Pic(X)\simeq\mathbb Z\langle H\rangle$ by \cite{BL}. From Theorem \ref{reduction}(3) $\mathcal{L}_x\subset \p^{n-1}$ is smooth irreducible and non-degenerate. Being defined scheme theoretically by $c\leq \frac{n-1}{2}$ equations, it is a complete intersection by the result of Faltings \cite{Faltings}. By the previous point $X$ is a complete intersection, too. 

Finally, part (4) follows from part (3) and Proposition~\ref{index}.
\end{proof}

\begin{Theorem}\label{HV} Assume that $X\subset \p^N$ is as in {\rm($*$)} and $X$ is quadratic. If $n=2c$ and $X$ is not a complete intersection, then it is projectively equivalent to one of the following:
\begin{enumerate}
\item [(a)] $\mathbb G(1,4)\subset\p^9$, or
\item [(b)] $S^{10}\subset\p^{15}$.
\end{enumerate}
\end{Theorem}
\begin{proof}
We need a refinement of Faltings' Theorem from \cite{Faltings}, due to Netsvetaev, see \cite {Net}.
Since we assumed that $X$ is not a complete intersection, we have $a\geq n-c=\frac{n}{2}\geq \frac{n-1}{2}$, so $\mathcal{L}_x\subset \p^{n-1}$ is smooth irreducible non-degenerate by Theorem~\ref{reduction}~(3). Moreover, by the previous theorem, $\mathcal{L}_x$ is not a complete intersection in $\p^{n-1}$, but it is scheme theoretically defined by $c$ independent quadratic equations. Netsvetaev's Theorem from \cite{Net} applied to $\mathcal{L}_x\subset \p^{n-1}$ gives the inequalities
$$\frac{3n-6}{4}\leq a <\frac{3n-5}{4} .$$
We may assume $n\geq 6$ (otherwise $\mathcal{L}_x$ is a complete intersection), so we deduce that $n=4r+2$ and $a=3r$, for a suitable $r$. If $r > 2$ (equivalently $n > 10$), we would have $a\geq 2(n-1-a)+1$ and $\mathcal{L}_x$ would be a complete intersection by Theorem~\ref{HaCo}~(3). In conclusion $n=6$ or $n=10$. In the first case $i(X)=a+2=5$ and we get case (a) by the classification of del Pezzo manifolds, see \cite{Fujita}. In the second case, $i(X)=8$, leading to case (b) by \cite{Mukai}.
\end{proof}

\begin{Remarks}
\begin{enumerate}
\item If $X\subset \p^{\frac{3n}{2}}$ is not a complete intersection, it is called a Hartshorne manifold. $\mathbb G(1,4)\subset\p^9$ and $S^{10}\subset\p^{15}$ are such examples, the first one being due to Hartshorne himself. So, for quadratic manifolds we proved not only that the (HC) holds, but also that the above examples are the only Hartshorne manifolds. This shows, once again, how mysterious the general case of the (HC) remains!
\item Working with local differential geometric techniques, as in \cite{GH, IL}, Landsberg \cite{Landsberg} proved that a (possibly singular) quadratic variety satisfying $n\geq 3c+b-1$ is a complete intersection, where $b=\dim(\Sing(X))$.

\end{enumerate}

\end{Remarks}




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