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\def\Pic{{\mathop{\rm Pic}\nolimits}}
\def\cod{{\mathop{\rm cod}\nolimits}}



\begin{document}

\title{The Title}

\author{The Author(s)}

\markboth{The Author(s)}{The Title}

\maketitle

\begin{abstract}
This is only a sample text for the use of {\tt degm.cls} with \LaTeX2e.
If you want a different numbering of Theorems, Remarks, etc.
you need to change the according definitions at the beginning of this file.
\end{abstract}

\begin{classification}
57M25
\end{classification}


\section{Introduction}

Let $X\subset {\mathbb P}^{n} $ be an $n$-dimensional connected
projective submanifold of projective space. Let $p : {\mathbb P}^N
\to {\mathbb P}^{N-q-1}$ denote the projection from a linear
${\mathbb P}^{n} q\subset {\mathbb P}^{n} N$. Assuming that
$X\not\subset {\mathbb P}^{n} q$ we have the induced rational
mapping $\psi:=p_X: X\to {\mathbb P}^{n} {N-q-1}$. This article
started as an attempt to understand the structure of this mapping
when $\psi$ has a lower dimensional image.

The special case when $Y$ is a point is very classical: $X$ is a
linear subspace of ${\mathbb P}^{n} N$. The case when $q=1$ and
$Y={\mathbb P}^{n} q={\mathbb P}^{n} 1$ was settled for surfaces
by the fourth author \cite{Laz}. Beyond this even the special case
when $q\ge 2$ and $Y={\mathbb P}^{n} q$ is open.

We have found it convenient to study a closely related question,
which includes many special cases including the case when the
center of the projection ${\mathbb P}^{n} q$ is contained in $X$.


\begin{remark} Let $Y$ be a proper connected
$k$-dimensional projective submanifold of an $n$-dimensional
projective manifold $X$. Assume that $k>0$. Let $L$ be a very
ample line bundle on $X$ such that $ L\otimes {\cal J}_Y$ is spanned by
global sections, where ${\cal J}_Y$ denotes the ideal sheaf of $Y$ in
$X$. Describe the structure of $(X,Y,L)$ under the additional
assumption that the image of $X$ under the mapping $\psi$
associated to $| L\otimes {\cal J}_Y|$ is lower
dimensional.
\end{remark}

\section
{Background material}\label{Background} We work over the complex
numbers ${\mathbb CJ}$. Through the paper we deal with projective
varieties $V$. We denote by ${\cal O}_V$ the structure sheaf of
$V$ and by $K_V$ the canonical bundle, for $V$ smooth. For any
coherent sheaf ${\cal F}$ on $V$, $h^i({\cal F})$ denotes the
complex dimension of $H^i(V,{\cal F})$.

Let ${\cal L}$ be a line bundle on $V$. The line bundle ${\cal L}$ is said to be {\em
numerically effective {\rm
(}nef}, for short) if ${\cal L} \cdot C \geq 0$ for all effective curves $C$ on
$V$. ${\cal L}$ is said to be
{\em big} if $\kappa ({\cal L}) = \dim V$, where $\kappa({\cal L})$ denotes the Kodaira
dimension of ${\cal L}$. If ${\cal L}$ is
nef then this is equivalent to $c_1({\cal L})^n > 0$, where $c_1({\cal L})$ is the
first Chern class of ${\cal L}$
and $n= \dim V$.


The following standard lemma is basic (see also \cite[(3.1.8)]{Book}).

\begin{lemma}\label{BasicLemma}Let $V$ be an irreducible
normal projective variety with $\Pic V\cong {\mathbb Z}$. Let
$g:V\to Z$ be a surjective morphism of $V$ to a projective variety
$Z$. Either $g$ is a finite morphism or $g(V)$ is a point. The
same conclusion holds for $V\cong {\mathbb P}^{n} n$ and any
holomorphic map to a compact complex space $Z$.
\end{lemma}

\proof Assume that $g$ is not finite and doesn't map $V$ to a
point. If $Z$ is projective, then the pullback of an ample line
bundle cannot be ample and thus we see that $\Pic V\not \cong
{\mathbb Z}$. Thus we can assume that $V\cong {\mathbb P}^{n} n$
and $Z$ is not necessarily projective.


Note that $\dim g({\mathbb P}^{n} n)=n$. If not let $F$ denote a
general fiber. Since it is smooth it would have trivial normal
bundle. This contradicts the ampleness of the tangent bundle of
${\mathbb P}^{n} n$.

Let $F$ denote a positive dimensional fiber. There is a complex
neighborhood $U$ of $F$ which maps generically one-to-one to a
Stein space. Since ${\mathbb P}^{n} n$ is homogeneous we have that
the translates of $F$ fill out an open set. Since the map $g_U$
must map these positive dimensional subspaces to points we have
the contradiction that $\dim_g(U)=\dim_g({\mathbb P}^{n} n)<n$.
\qed


\begin{definition} Let $f$ be a form in a domain $D$. We say that
$f$ is quasibounded if there exists a bounded plurisubharmonic
function, $\phi$ , in $D$ such that $|f|_{\delta\phi}$ is uniformly
bounded in $D$.
\end{definition}

\begin{lemma}[A. Smith]\label{HartRev}
Assume that Hartshorne's conjecture {\rm \cite{Book}}, that any
connected nondegenerate $n$-dimensional smooth submanifold
$X\subset {\mathbb P}^{n} m$ is a complete intersection if
$n>\frac{2}{3}m$, is true. Then each vector bundle ${\cal E}$ on
${\mathbb P}^{n} m$ of rank $r<\frac{m}{3}$ splits into a direct
sum of line bundles.
\end{lemma}


\begin{corollary}\label{Pk} Let $X$ be a compact subset of ${\mathbb C}^n$.  Then the following
are equivalent:
\begin{enumerate}
\item[\rm (1)] $J_q(X)=X$.
\begin{enumerate}
\item[\rm (a)] $C(X)=P_q(X)$.
\item[\rm (b)] For any $z\in X$, there exists an $r>0$ such that $X_r=X\cap \overline{B(z, r)}$.
\end{enumerate}
\item[\rm (2)] For any $M>0$, there exists a function
$\lambda$ smooth in a neighborhood of $X$ such that\/ $0<\lambda (z)<1$ and
$H_q(\lambda)(z, u)\ge M |u|^2$ for $z\in X$ and $u\in \Lambda^{(0, q)}_z$.
\end{enumerate}
\end{corollary}






\begin{theorem}\label{threer1}
Let $X$ be a smooth threefold, and let $L$ a very ample line
bundle on $X$. Assume that $(X,L,P)$ is a ${\mathbb P}^{n}
2$-degenerate triple. If $s=1$ and ${\mathfrak t}:=\deg \phi_P\ge
2 $, then $t=4$. In this case $X$ is the blowing up at one point
of the complete intersection of three quadrics in ${\mathbb
P}^{n}$.
\end{theorem}

\proof If $s=1$ then by Proposition \ref{HartRev} we see that $L^3=7$. Note
that we use the
classification of degree $7$ manifolds given in \cite{Io}. We can assume
that $K_X+2L$ is nef and big. Thus quadric fibrations over curves and
scrolls over curves and
surfaces are ruled out. By using the degree $7$ classification, two
possibilities remain.

\begin{enumerate}

\item $X$ is the blowing up at one point, $\pi:X\to X'$, of the complete
intersection $X'$ of three quadrics in ${\mathbb P}^{n} 6$, with
$\pi$ the first reduction map; or

\item there
exists a morphism
\begin{equation}
\rho:X\to C
\end{equation}
 of $X$ to a curve $C$ given by the complete
linear system $|m(K_X+L)|$ for $m\gg 0$.
\end{enumerate}

In the first case we know from \cite{Io} that $L$ embeds $X$ into
${\mathbb P}^{n} 5$. This $X$ contains the positive dimensional
fiber of $\pi$ and thus since projection from this linear
${\mathbb P}^{n} 2$ must map to ${\mathbb P}^{n} 2$ we conclude
that this is an example with $s=1$. Let $f\cong{\mathbb P}^{n} 1$
be a fiber of $\phi:X\to Z$. To see what $t$ is, note that $K_X+L$
being nef yields $t=L\cdot f\geq -K_X\cdot f=2$. By (2.1)
  we know that
$P$ coincides with the exceptional divisor of $\pi$. Moreover,
$-K_{X'}\cong{\cal O}_{X'}(1)=L'$, the
polarization of the first reduction $X'$, which satisfies the condition
$L\cong\pi^*L'-P$. Then
$$K_X\cong\pi^*K_{X'}+2P\cong -L-P+2P=-L.$$

In the second case $\rho(P)$ must be a point by Lemma \ref{BasicLemma} and
therefore $(K_X+L)_P\cong{\cal O}_P$. Since
$(K_X+L)_P\cong{\cal O}_P(s-2)$ we get the contradiction
$s=2$.\qed




\frenchspacing
\begin{thebibliography}{9}

\bibitem{Book} M. C. Beltrametti and A. J. Sommese,  The Adjunction
Theory of Complex Projective Varieties, de Gruyter Exp. Math. 16,
Walter de Gruyter, Berlin--New York 1995.

\bibitem{Io} P. Ionescu, Embedded projective varieties of small invariants,
in:  Proceedings of the
1982 Week of Algebraic Geometry (L. B\v{a}descu and D.
Popescu, eds.),  Lecture Notes in Math. 1056, Springer-Verlag, New York 1984,
142--187.

\bibitem{Laz} R. Lazarsfeld, A sharp Castelnuovo bound for smooth surfaces,
Duke Math. J. 55 (1987), 423--429.

\bibitem{Ran} Z. Ran, Local differential geometry and generic projections
of threefolds, J. Differential Geom. 32 (1990), 131--137.

\bibitem{Zar1} O. Zariski, Algebraic Surfaces, with appendices by
S. S. Abhyankar, J. Lipman, and D. Mumford, Ergeb.  Math.  Grenzgeb. (2)
61, Springer-Verlag, Berlin 1971.

\end{thebibliography}




\end{document}
