\documentclass[nomath,noamsfonts]{tran-l} \newtheorem{Theorem}{Theorem}[section] \newtheorem{Lemma}[Theorem]{Lemma} \newtheorem{Proposition}[Theorem]{Proposition} \newtheorem{Corollary}[Theorem]{Corollary} \theoremstyle{definition} \newtheorem{Definition}[Theorem]{Definition} \theoremstyle{remark} \newtheorem{remark}[Theorem]{Remark} \theoremstyle{remark} \newtheorem{claim}[Theorem]{Claim} \theoremstyle{remark} \newtheorem{case}[Theorem]{Case} \numberwithin{equation}{section} \newcommand{\E}{E} \newcommand{\taut}{{\xi}_E} \newcommand{\Proj}{{\bf P}} \newcommand{\Segno} {\refstepcounter{Theorem} \par \noindent{\sl (\theTheorem) }} \newcommand{\mapright}[1]{\smash{\mathop{\longrightarrow}\limits^{#1}}} \newcommand{\mapdown}[1]{\Big \downarrow \rlap{$\vcenter{\hbox{$\scriptstyle#1$}}$}} % these are frequently used bold letters and calligraphic \newcommand{\Ol}{{\mathcal O}} \newcommand{\D}{{\mathcal D}} \newcommand{\F}{{\mathcal F}} \newcommand{\I}{{\mathcal I}} \newcommand{\Q}{{\bf Q}} \newcommand{\N}{{\bf N}} \newcommand{\Z}{{\bf Z}} \newcommand{\C}{{\bf C}} \newcommand{\G}{{\Gamma}} \newcommand{\R}{{\bf R}} % now morphisms \newcommand{\f}{\varphi} \newcommand{\ra}{\rightarrow} \newcommand{\iso}{\simeq} %and projectivisations \newcommand{\PE}{{\bf P}({\mathcal E})} \begin{document} \title[Contractions polarized by vector bundles]{Contractions on a manifold polarized by an ample vector bundle} % author one information \author{Marco Andreatta} \address{Dipartimento di Matematica,Universit\'a di Trento, 38050 Povo (TN), Italia} \email{andreatt@science.unitn.it} % author two information \author{Massimiliano Mella} \address{Dipartimento di Matematica,Universit\'a di Trento, 38050 Povo (TN), Italia} \email{mella@science.unitn.it} \subjclass{Primary 14E30, 14J40; Secondary 14C20,14J45} \keywords{Vector bundle, contraction, extremal ray} \begin{abstract} A complex manifold $X$ of dimension $n$ together with an ample vector bundle $E$ on it will be called a {\sf generalized polarized variety}. The adjoint bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. We study the positivity (the nefness or ampleness) of the adjoint bundle in the case $r := rank (E) = (n-2)$. If $r\geq (n-1)$ this was previously done in a series of paper by Ye-Zhang, Fujita, Andreatta-Ballico-Wisniewski. \par If $K_X+detE$ is nef then, by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map $\pi :X \longrightarrow W$ from $X$ onto a normal projective variety $W$ with connected fiber and such that $K_X + det(E) = \pi^*H$, for some ample line bundle $H$ on $W$. We describe those contractions for which $dimF \leq (r-1)$. We extend this result to the case in which $X$ has log terminal singualarities. In particular this gives the Mukai's conjecture1 for singular varieties. We consider also the case in which $dimF = r$ for every fibers and $\pi$ is birational. \end{abstract} \maketitle \section*{Introduction} An algebraic variety $X$ of dimension $n$ (over the complex field) together with an ample vector bundle $E$ on it will be called a {\sf generalized polarized variety}. The adjoint bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. Problems concerning adjoint bundles have drawn a lot of attention to algebraic geometer: the classical case is when $E$ is a (direct sum of) line bundle (polarized variety), while the generalized case was motivated by the solution of Hartshorne-Frankel conjecture by Mori ( \cite{Mo}) and by consequent conjectures of Mukai (\cite{Mu}). \par A first point of view is to study the positivity (the nefness or ampleness) of the adjoint line bundle in the case $r = rank (E)$ is about $n = dim X$. This was done in a sequel of papers for $r\geq n-1$ and for smooth manifold $X$ ([Ye-Zhang], [Fujita], [Andreatta-Ballico-Wisniewski]). In this paper we want to discuss the next case, namely when $rank (E) = n-2$, with $X$ smooth; we obtain a complete answer which is described in the theorem (\ref{main}). This is divided in three cases, namely when $K_X + det(E)$ is not nef, when it is nef and not big and finally when it is nef and big but not ample. If $n=3$ a complete picture is already contained in the famous paper of Mori (\cite{Mo1}), while the particular case in which $E = \oplus^{n-2} (L)$ with $L$ a line bundle was also studied (\cite{Fu1}, \cite{So}; in the singular case see \cite{An}). The part 1 of the theorem was proved (in a slightly weaker form) by Zhang (\cite{Zh}) and, in the case $E$ is spanned by global sections, by Wisniewski (\cite{Wi2}). \par Another point of view can be the following: let $(X,E)$ be a generalized polarized variety with $X$ smooth and $rankE=r$. If $K_X + det(E)$ is nef, then by the Kawamata-Shokurov base point free theorem it supports a contraction (see (1.2)); i.e. there exists a map $\pi :X \ra W$ from $X$ onto a normal projective variety $W$ with connected fiber and such that $K_X + det(E) = \pi^*H$ for some ample line bundle $H$ on $W$. It is not difficult to see that, for every fiber $F$ of $\pi$, we have $dimF \geq (r-1)$, equality holds only if $dimX > dimW$. In the paper we study the "border" cases: we assume that $dimF = (r-1)$ for every fibers and we prove that $X$ has a $\Proj^r$-bundle structure given by $\pi$ (theorem (3.2)). We consider also the case in which $dimF = r$ for every fibers and $\pi$ is birational, proving that $W$ is smooth and that $\pi$ is a blow-up of a smooth subvariety (theorem (3.1)). This point of view was discussed in the case $E = \oplus^r L$ in the paper [A-W]. \par Finally in the section (4) we extend the theorem (3.2) to the singular case, namely for projective variety $X$ with log-terminal singularities. In particular this gives the Mukai's conjecture 1 for singular varieties. \par During the preparation of this paper we were partially supported by the MURST and GNSAGA. We would like to thank also the Max-Planck-Institute f\"ur Mathematik in Bonn and Warwick University for support and hospitality. \section{Notations and generalities} \Segno We use the standard notation from algebraic geometry. In particular it is compatible with that of [K-M-M] to which we refer constantly. We just explain some special definitions and propositions used frequently. In particular in this paper $X$ will always stand for a smooth complex projective variety of dimension $n$. Let $Div(X)$ the group of Cartier divisors on $X$; denote by $K_X$ the {\sf canonical divisor} of $X$, an element of $Div(X)$ such that $\Ol_{X}(K_X) = \Omega^n_{X}$. Let $N_1(X)=\frac{\{1-cycles\}}{\equiv}\otimes \R$, $N^1(X)= \frac{\{divisors\}}{\equiv}\otimes \R$ and $\overline {}=\overline{\{\mbox{effective 1-cycles}\}}$; the last is a closed cone in $N_1(X)$. Let also $\rho(X)=dim_{\R}N^1(X)$. Suppose that $K_X$ is not nef, that is there exists an effective curve $C$ such that $K_X\cdot C<0$. \begin{Theorem}\cite{KMM} Let $X$ as above and $H$ a nef Cartier divisor such that $F:= H^{\bot} \cap \overline {} \setminus \{0\}$ is entirely contained in the set $\{Z\in N_1(X) :K_X\cdot Z<0\}$, where $H^{\bot} = \{Z:H\cdot Z=0\}$. Then there exists a projective morphism $\f:X\ra W$ from $X$ onto a normal variety $W$ with the following properties: \begin{itemize} \item[{i})] For an irreducible curve $C$ in $X$, $\f(C)$ is a point if and only if $H.C = 0$, if and only if $cl(C) \in F$. \item[{ii})] $\f$ has only connected fibers \item[{iii})] $H = \f^*(A)$ for some ample divisor $A$ on $W$. \item[{iv})] The image $\f^* :Pic(W) \ra Pic(X)$ coincides with $\{D \in Pic(X): D.C = 0 \mbox{ \rm for all } C \in F\}.$ \end{itemize} \label{contractionth} \end{Theorem} \begin{Definition} (\cite{KMM}, definition 3-2-3). Using the notation of the above theorem, the map $\f$ is called a {\sf contraction} (or an {\sf extremal contraction}); the set $F$ is an {\sf extremal face}, while the Cartier divisor $H$ is a {\sf supporting divisor} for the map $\f$ (or the face $F$). If $dim_{\R}F = 1$ the face $F$ is called an {\sf extremal ray}, while $\f$ is called an {\sf elementary contraction}. \end{Definition} \begin{remark} We have also (\cite{Mo1}) that if $X$ has an extremal ray $R$ then there exists a rational curve $C$ on $X$ such that $0< -K_X \cdot C\leq n+1$ and $R=R[C]:=\{D\in : D\equiv \lambda C, \lambda\in \R^+\}$. Such a curve is called an {\sf extremal curve}. \end{remark} \begin{remark}\label{biraz} Let $\pi:X\ra V$ denote a contraction of an extremal face $F$, supported by $H=\pi^*A$. Let $R$ be an extremal ray in $F$ and $\rho:X\ra W$ the contraction of $R$. Then $\pi$ factors trough $\rho$ (this is because $\pi^*A\cdot R=0$). \end{remark} \begin{Definition} To an extremal ray $R$ we can associate: \begin{itemize} \item[{i})] its {\sf length} $l(R):=min\{ -K_X\cdot C;$ for $C$ rational curve and $C\in R\}$ \item[{ii})] the {\sf locus} $E(R):=\{$the locus of the curves whose numerical classes are in $R\}\subset X$. \end{itemize} A rational curve $C$ in $R$ such that $-K_X\cdot C=l(R)$ will be called a minimal curve \label{mindef} \end{Definition} It is usual to divide the elementary contractions associated to an extremal ray $R$ in three types, according to the dimension of $E(R)$ as it follows. \begin{Definition} We say that $\f$ is of {\sf fiber type}, respectively {\sf divisorial type}, resp. {\sf flipping type}, if $dim E(R) = n$, resp. $n-1$, resp. $< n-1$. Moreover an extremal ray is said not nef if there exists an effective $D\in Div(X)$ such that $D\cdot C<0$. \end{Definition} The following very useful inequality was proved in \cite{Io} and \cite{Wi3}. \begin{Proposition} Let $\varphi$ the contraction of an extremal ray $R$, $E^{\prime}(R)$ be any irreducible component of the exceptional locus and $d$ the dimension of a fiber of the contraction restricted to $E^{\prime}(R)$. Then $$ dim E^{\prime}(R)+d\geq n+l(R)-1.$$ \label{diswis} \end{Proposition} \Segno Actually it is very useful to understand when a contraction is elementary or in other words when the locus of two distinct extremal rays are disjoint. For this we will use in this paper the following results. \begin{Proposition}\cite[Corollary 0.6.1]{BS} Let $R_1$ and $R_2$ two distinct not nef extremal rays such that $l(R_1)+l(R_2)>n$. Then $E(R_1)$ and $E(R_2)$ are disjoint. \label{birelementare} \end{Proposition} Something can be said also if $l(R_1)+l(R_2)=n$: \begin{Proposition}\cite[Theorem 2.4]{Fu3} Let $\pi:X\ra V$ a birational contraction of a face $F$; suppose $n\geq 4$ and $l(R_i)\geq n-2$, for $R_i$ extremal rays in $F$. Then the exceptional loci corresponding to different extremal rays, are disjoint with each other. \label{n=4} \end{Proposition} \begin{Proposition}\cite{ABW1} Let $\pi:X\ra W$ be a contraction of a face such that $dimX > dim W$. Suppose that for every rational curve $C$ in a general fiber of $\pi$ we have $-K_X\cdot C\geq (n+1)/2$. Then $\pi$ is an elementary contraction except if \begin{itemize} \item[a)] $-K_X\cdot C=(n+2)/2$ for some rational curve $C$ on $X$, $W$ is a point, $X$ is a Fano manifold of pseudoindex $(n+2)/2$ and $\rho(X)=2$ \item[b)] $-K_X\cdot C=(n+1)/2$ for some rational curve $C$, and $dim$W$\leq 1$ \end{itemize} \label{fibelementare} \end{Proposition} Finally the following definitions are used in the main theorem in section 5: \begin{Definition} Let $L$ be an ample line bundle on $X$. The pair $(X,L)$ is called a scroll (respectively a quadric fibration, respectively a del Pezzo fibration) over a normal variety $Y$ of dimension $m$ if there exists a surjective morphism with connected fibers $\phi: X \ra Y$ such that $$K_X+(n-m+1)L \approx p^*{\mathcal L}$$ (respectively $K_X+(n-m)L \approx p^*{\mathcal L}$, respectively $K_X+(n-m-1)L \approx p^*{\mathcal L}$) for some ample line bundle ${\mathcal L}$ on $Y$. $X$ is called a classical scroll (respectively quadric bundle) over a projective variety $Y$ of dimension $r$ if there exists a surjective morphism $\phi : X\ra Y$ such that every fiber is isomorphic to $\Proj^{n-r}$ (respectively to a quadric in $\Proj^{n-r+1}$) and if there exists a vector bundle $E$ of rank $n-r+1$ (respectively of rank $n-r+2$) on $Y$ such that $X\simeq \Proj(E)$ (respectively exists an embedding of $X$ as a subvariety of $\Proj(E)$). \end{Definition} \section{A technical construction} \label{tech} Let $E$ be a vector bundle of rank $r$ on $X$ and assume that $E$ is ample (in Hartshorne's sense). \begin{remark} Let $f:{\bf P}^1\rightarrow X$ be a non constant map, and $C=f({\bf P}^1)$, \label{mha} then $detE\cdot C\geq r$. \par In particular if there exists a curve $C$ such that $(K_X+detE).C \leq 0$ (for instance if $(K_X+detE)$ is not nef) then there exists an extremal ray $R$ such that $l(R) \geq r$. \end{remark} \Segno \label{sopra} Let $Y=\Proj(E)$ be the associated projective space bundle, $p:Y \rightarrow X$ the natural map onto $X$ and $\taut$ the tautological bundle of $Y$. Then we have the formula for the canonical bundle $K_Y=p^*(K_X+detE)-r\taut$. Note that $p$ is an elementary contraction. Assume that $K_X+detE$ is nef but not ample and that it is the supporting divisor of an elementary contraction $\pi:X\rightarrow W$, let $R$ be the associated extremal ray. Then $\rho(Y/W) = 2$ and $-K_Y$ is $\pi\circ p$-ample. By the relative Mori theory over $W$ we have that there exists a ray on $NE(Y/W)$, say $R_1$, of length $\geq r$, not contracted by $p$, and a relative elementary contraction $\f:Y\ra V$. We have thus the following commutative diagram. \begin{equation} \label{dia1} \matrix{{\bf P}(E)=Y&\mapright\varphi&V\cr \mapdown{p}&&\mapdown\psi\cr X&\mapright\pi&W} \end{equation} where $\varphi$ and $\psi$ are elementary contractions. Let $w\in W$ and let $F(\pi)_w$ be an irreducible component of $\pi^{-1}(w)$; choose also $v$ in $\psi^{-1}(w)$ and let $F(\varphi)_v$ be an irreducible component of $\varphi ^{-1}(v)$ such that $p(F(\varphi)_v) \cap F(\pi)_w \not= \emptyset$; then, by the commutativity of the diagram, $p(F(\varphi)_v) \subset F(\pi)_w$. Since $p$ and $\f$ are elementary contractions of different extremal rays we have that $dim(F(\varphi)\cap F(p))=0$, that is a curve which is contracted by $\varphi$ cannot be contracted by $p$. In particular this implies that $dim (p(F(\varphi)_v)) = dim F(\varphi)_v$; therefore $$dimF(\varphi)_v\leq dimF(\pi)_w.$$ \begin{remark} If $dimF(\varphi)_v=dimF(\pi)_w$, then $dim F(\psi)_w:=dim(\psi^{-1}(w)) = r-1$; if this holds for every $w \in W$ then $\psi$ is equidimensional. \label{fdim} \end{remark} \begin{proof} Let $Y_w$ be an irreducible component of $p^{-1}\pi^{-1}(w)$ such that $\varphi (Y_w) = F(\psi)_w$. Then $dim F(\psi)_w = dim Y_w - dimF(\varphi)_v = dim Y_w - dimF(\pi)_w = dimF(p) = r-1$. \end{proof} \Segno {\bf Slicing techniques} \label{adj} Let $H = \varphi ^*(A)$ be a supporting divisor for $\varphi$ such that the linear system $|H|$ is base point free. We assume as in (\ref{sopra}) that $( K_X + detE)$ is nef and we refer to the diagram (\ref{dia1}). The divisor $K_Y+r\taut =p^*( K_X + detE)$ is nef on $Y$ and therefore $m(K_Y+r\taut+aH)$, for $m\gg 0$, $a\in{\bf N}$, is also a good supporting divisor for $\varphi$. Let $Z$ be a smooth n-fold obtained by intersecting $r-1$ general divisor from the linear system H, i.e. $Z = H_1\cap \dots \cap H_{r-1}$ (this is what we call a {\sf slicing}); let $H_i = \varphi^{-1} A_i$. Note that the map ${\varphi}^{\prime}=\varphi_{|Z}$ is supported by $|m(K_Y+r\taut+a\varphi^*A)_{|Z}|$, hence, by adjunction, it is supported by $K_Z+rL$, where $L={\taut}_{|Z}$. Let $p^{\prime}=p_{|Z}$; by construction $p^{\prime}$ is finite. If $T$ is (the normalization of) $\varphi (Z)$ and $\psi^{\prime} :T \ra W$ is the map obtained restricting $\psi$ then we have from (\ref{dia1}) the following diagram \begin{equation} \label{dia2} \matrix{Z&\mapright{\varphi\prime}&T\cr \mapdown{p\prime}&&\mapdown{\psi\prime}\cr X&\mapright\pi&W} \end{equation} In general the map $\f^{\prime}$ is well understood (for instance in the case $r = n-2$ see the results in \cite{Fu1} or in \cite{An}). The goal is to "transfer" the information that we have on $\f^{\prime}$ to the map $\pi$. The following proposition is an example. We refer to the diagrams and notations of the above sections; in particular $\pi : X \ra W$ is the elementary contraction of the ray $R$ supported by $K_X+detE$. Therefore $l(R) \geq r$ and, by the proposition {\ref{diswis}} we have $$ dim E^{\prime}(R)+d\geq n+r-1$$ where $E^{\prime}(R)$ is an irreducible component of the exceptional locus and $d = dimF(\pi)$. \begin{Proposition} Assume that for every non trivial fiber we have $dimF(\varphi)=dimF(\pi) = k$. Assume also that $l(R)=r$ and that for all fiber of $\f$ $(F(\f),\taut{}_{|F(\f)})\simeq (\Proj^k,\Ol(1))$. Then $W$ has the same singularities of $T$. \label{fujita} \end{Proposition} \begin{remark} The above proposition was proved in the case in which $\f$ is birational and $k = r$ in \cite{ABW2}. \end{remark} \begin{proof} Let $w\in W$; by hypothesis and by the remark (\ref{fdim}) any irreducible component $F_i$ of a fiber $F(\psi)_w$ is of dimension $r-1$. This implies also that $F_i=\f(F(p))$ for some fiber of $p$. \begin{Lemma} There exists a fiber $F(p)_x$ such that $\f_{|F(p)_x}:F(p)_x\ra F_i$ is of degree 1, that is $\f_{|F(p)_x}$ is set theoretically birational. \end{Lemma} \begin{proof} For every $v \in V$ we have that $\f_{|F(p)_x}^{-1}(v) = F(p)_x \cap F(\f)_v$; therefore the lemma follows if we can prove that, for a general $v\in V$, with $\psi(v) =w$, $p_{|F(\f)_v}:F(\f)_v\ra F(\pi)_w$ is set theoretically birational. \par We will need the following claim. \begin{claim} Let $l$ be a line in $F(\f)\simeq \Proj^k$; then $-p^*K_X\cdot l = r$. \end{claim} \begin{proof}[ Proof of the claim] Let $C$ a minimal curve in the ray $R$ (see (\ref{mindef})); let $\nu:\Proj^1\ra C$ be its normalization. Thus $\nu^*E_{|C}=\oplus^r\Ol(1)$ and therefore $Y_C=\Proj( \nu^*E_{C})=\Proj^1\times\Proj^{r-1}$. Let $\tilde{\nu}:Y_C\ra Y$ the map induced by $\nu$ and let $\tilde{l}$ be a section of $\sigma: Y_C \ra \Proj^1$; note that $\nu \sigma: \tilde{l}\ra C$ is birational. Note also that $\tilde{\nu}^*\taut$ is the tautological bundle for $Y_C$; thus $1=\tilde{\nu}^*\taut\cdot \tilde{l} =\taut\cdot \tilde{\nu}_*\tilde{l}$, hence $\tilde{\nu}_*\tilde{l}=l$. Therefore $p_*l = C$ and $-p^*K_X\cdot l=-K_X\cdot p_*l=-K_X\cdot C=r$. \end{proof} Let $R$ be the ramification divisor of $p':Z \ra X$ defined by the formula $$K_Z=p^{\prime*}K_X+R.$$ Let $l$ be a line in $F(\f) = \Proj^k \subset Z$; on one side we have that $-K_Z\cdot l=r$, on the other, by the above claim, $p^*K_X\cdot l = r$. Therefore $R\cdot l=0$. Thus either $F(\f)\subset R$ or $F(\f)\cap R= \emptyset$, we want to prove that the latter is the case. \begin{Lemma} For a general choice of $Z$ the ramification divisor $R$ does not contain $F(\f) = \Proj^k \subset Z$; therefore $F(\f) \cap R = \emptyset$ \end{Lemma} \begin{proof} It is enough to prove that there exists an $x \in F(\pi)_w$ such that $p^{-1}(x) \cap Z = d$ distinct points, where $d = deg (p': Z \ra X)$. Observe that this is true for every $x_1\in X$ outside the branch locus and $d=\f^*A^{r-1}\cdot F(p)_{x_1}=\f^*A^{r-1} \cdot F(p)_x$, where $Z=\f^*A_1\cap\ldots\cap\f^*A_{r-1}$ and $A_i\in|A|$. Moreover $p^{-1}(x) \cap Z =\cup_i p^{-1}(x) \cap F(\f)_{v_i}$, where the union is taken over all $v_i\in T\cap F_i$. Since $\f_{|F(p)_x}: F(p)_x\ra F_i$ is generically unramified then choosing generic sections $A_i\in |A|$ yields that $p^{-1}(x) \cap F(\f)_{v_i}$ is a reduced cycle of length $d_i$ for any $i$ and $\sum_i d_i=d$. Hence $F(\f)\cap R=\emptyset$. \end{proof} The following exact sequence $$\I/\I^2\ra \Omega_{Z/X}\otimes \Ol_{F(\f)}\ra \Omega_{F(\f)/X}\ra 0,$$ yields that also $p_{|F(\f)}:\Proj^k\ra F(\pi)$ is unramified. Let $f:\tilde{F}\ra F(\pi)$ the normalization and $g:\Proj^k\ra \tilde{F}$ the map induced by $p$; then $g$ is unramified and $\tilde{F}$ is smooth by Zarisky Main Theorem. Therefore $\tilde{F}\simeq \Proj^k$ by Lazarsfeld result and $g$ is an isomorphism; thus $p_{|F(\f)}$ is of degree 1. \end{proof} Let $\f_{|F(p)}:F(p)\ra F_i$ be as in the lemma, that is $\f_{|F(p)}$ is set theoretically birational Let us follow an argument as in \cite[Lemma 2.12]{Fu1}. We can assume that the divisor $A$ is very ample. Using Bertini's theorem we choose $r-1$ divisors $A_i \in |A|$ as above such that, if $T = {\bigcap_{i}} A_i$, then $T \cap \psi^{-1}(w)_{red}= N$ is a reduced 0-cycle and $Z = H_1\cap \dots \cap H_{r-1}$ is a smooth n-fold, where $H_i = \varphi^{-1} A_i$. Moreover the number of points in $N$ is given by $A^{r-1}\cdot \psi^{-1}(w)_{red}=\sum_i A^{r-1}\cdot F_i=\sum_i d_i$. Note that, by projection formula, we have $A^{r-1}\cdot F_i= \varphi^*A^{r-1}\cdot F(p)$; here we use the fact that the map $\f_{|F(p)}$ is set theoretically birational. Moreover, since $p$ is a projective bundle, the last number is constant i.e. $\varphi^*A^{r-1}\cdot F(p) = d$ for all fiber $F(p)$, that is the $d_i$'s are constant. Using that $\psi^{\prime}:=\psi_{|T}: T \ra W$ is proper and finite over $w$, we now take a small enough neighborhood $U$ of $w$, in the metric topology, such that any connected component $U_{\lambda}$ of $\psi^{-1}(U)\cap T$ meets $\psi^{-1}(w)$ in a single point. Let $\psi_{\lambda}$ the restriction of $\psi$ at $U_{\lambda}$ and $m_{\lambda}$ its degree. Then $deg{\psi}^{\prime}=\sum m_{\lambda}\geq \sum_i d_i = \sum_i d$ and equality holds if and only if $\psi$ is not ramified at $w$ (remember that $\sum_i d_i$ is the number of $U_{\lambda}$). The generic $F(\psi)_w$ is irreducible and generically reduced. Note that we can choose $\tilde{w}\in W$ such that $\psi^{-1}(\tilde{w})= \varphi(F(p))$ and $deg{\psi}^{\prime}=A^{r-1}\cdot\psi^{-1}(\tilde{w})$, the latter is possible by the choice of generic sections of $|A|$. Hence, by projection formula $deg\psi^{\prime}= A^{r-1}\cdot \psi^{-1}(\tilde{w})= \varphi^*A^{r-1}\cdot F(p)=d$, that is $m_{\lambda}=1$ and the fibers are irreducible. Since $W$ is normal we can conclude, by Zariski's Main theorem, that $W$ has the same singularities as $T$. \end{proof} \begin{Corollary} In the hypothesis of the above proposition assume also that either $\f$ is birational and $k= r$, or that $\f$ is of fiber type and $k = (r-1)$. Then $W$ is smooth. \label{smooth} \end{Corollary} \begin{proof} \cite[Theorem 4.1]{AW} applies to the map $\f$ and gives that $T$ is smooth and $\f$ satisfies the hypothesis of proposition (\ref{fujita})(for the fiber type case it is actually a theorem in \cite{Fu1}). Thus by proposition (\ref{fujita}) also $W$ is smooth. \end{proof} \section{Some general applications} As an application of the above construction we will prove the following proposition; the case $r = n-1$ was proved in \cite {ABW2}. \begin{Proposition} Let $X$ be a smooth projective complex variety and $E$ be an ample vector bundle of rank $r$ on $X$. Assume that $K_X+detE$ is nef and big but not ample and let $\pi:X\ra W$ be the contraction supported by $K_X+detE$. Assume also that $\pi$ is a divisorial elementary contraction, with exceptional divisor $D$, and that $dim F\leq r$ for all fibers $F$. Then $W$ is smooth, $\pi$ is the blow up of a smooth subvariety $B: = \pi (D)$ and $E =\pi^*E^{\prime}\otimes[-D]$, for some ample $E^{\prime}$ on $W$. \label{bd} \end{Proposition} \begin{proof} In the previous section (\ref{smooth}) we have proved that $W$ is smooth. Therefore $\pi$ is a birational morphism between smooth varieties with exceptional locus a prime divisor and with equidimensional non trivial fibers; by \cite[Corollary 4.11]{AW} this implies that $\pi$ is a blow up of a smooth subvariety in $W$. We want to show that $E =\pi^*E^{\prime}\otimes[-D]$. Let $D_1$ be the exceptional divisor of $\varphi$; first we claim that $\taut+D_1$ is a good supporting divisor for $\f$. Let $C_1$ be a minimal curve in the ray $R_1$ (see (\ref{mindef})), contracted by $\f$; we have that $\taut \cdot C_1 = 1$. Observe that $(\taut+D_1)\cdot C_1=0$, while $(\taut+D_1)\cdot C>0$ for any curve $C$ with $\f(C)\not= pt$ (in fact $\taut$ is ample and $D_1\cdot C\geq 0$ for such a curve). Thus $\taut+D_1=\f^*A$ for some ample $A\in Pic(V)$; moreover using the projection formula $A\cdot l=1$, for any line $l$ in the fiber of $\psi$. Hence by Grauert's theorem $V=\Proj(E^{\prime})$ for some ample vector bundle $E^{\prime}$ on $W$. This yields, by the commutativity of diagram (1), $E\otimes D=p_*(\taut+D_1)=p_*\f^*A=\pi^*\psi_*A=\pi^*E^{\prime}$. \end{proof} Similarly ,for the fiber type case, we have the following. \begin{Theorem} Let $X$ be a smooth projective complex variety and $E$ be an ample vector bundle of rank $r$ on $X$. Assume that $K_X+detE$ is nef and let $\pi:X\ra W$ be the contraction supported by $K_X+detE$. Assume that $r\geq (n+1)/2$ and $dim F\leq r-1$ for any fiber $F$ of $\pi$. Then $\pi$ is a fiber type contraction, $W$ is smooth, for any fiber $F\simeq \Proj^{r-1}$ and $E_{|F}=\oplus^r\Ol(1)$. \label{relscroll} \end{Theorem} \begin{proof} Note that, by proposition (\ref{diswis}), $\pi$ is a contraction of fiber type and all the fibers have dimension $r-1$. Moreover the contraction is elementary, by proposition (\ref{fibelementare}). By corollary (\ref{smooth}) $W$ is smooth. We want to use an inductive argument to prove the theorem. If $dim W=0$ then this is Mukai's conjecture 1 which was proved by Peternell, Koll\'ar, Ye-Zhang (see for instance \cite{YZ}). Let the theorem be true for dimension $m-1$. Note that the locus over which the fiber is not $\Proj^{r-1}$ is discrete. In fact take a general hyperplane section $A$ of $W$, and $X^{\prime}=\pi^{-1}(A)$ then $\pi_{|X^{\prime}}:X^{\prime}\ra A$ is again a contraction supported by $K_{X^{\prime}}+det E_{|X^{\prime}}$, such that $r\geq ((n-1)+1)/2$. Thus by induction $A$ is smooth and all fibers over $A$ are $\Proj^{r-1}$. \par Let $U$ be an open disk in the complex topology such that $U\cap SingW=\{0\}$. Then by lemma below \ref{scroll} we obtain locally, in the complex topology, a $\pi$-ample line bundle $L$ such that restricted to the general fiber is $\Ol(1)$. Thus, as in \cite[Prop. 2.12]{Fu1}, we can prove that all the fibers are $\Proj^{r-1}$. \end{proof} \begin{Lemma} \label{scroll} Let $X$ be a complex manifold and $(W,0)$ an analityc germ such that $W\setminus \{0\}\simeq \Delta^m\setminus \{0\}$. Assume we have an holomorphic map $\pi:X\ra W$ with $-K_X$ $\pi$-ample; assume also that $F\simeq \Proj^r$ for all fibers of $\pi$, $F\not= F_0=\pi^{-1}(0)$, and that $codim F_0\geq 2$. Then there exists a line bundle $L$ on $X$ such that $L$ is $\pi$-ample and $L_{|F}=\Ol(1)$. \end{Lemma} \begin{proof}(see also \cite[pag. 338-339]{ABW2}) Let $W^*=W\setminus \{0\}$ and $X^*=X\setminus F_0$. By abuse of notation call $\pi=\pi_{|X^*}:X^*\ra W^*$; it follows immediately that $R^1\pi_*\Z_{X^*}=0$ and $R^2\pi_*\Z_{X^*}=\Z$. Using the Leray spectral sequence, we have that: $$ E^{0,2}_2= \Z\mbox{ and } E^{p,1}_2= 0 \mbox{ for any p.}$$ Therefore $d_2:E^{0,2}_2\ra E^{2,1}_2$ is the zero map and moreover we have the following exact sequence $$0\ra E^{0,2}_{\infty}\ra E^{0,2}_2\stackrel{d_3}{\ra} E^{3,0}_2,$$ since the only non zero map from $ E^{0,2}_2$ is $d_3$ and hence $E^{0,2}_{\infty}=kerd_3$. On the other hand we have also, in a natural way, a surjective map $H^2(X^*,\Z)\ra E^{0,2}_{\infty}\ra 0$. Thus we get the following exact sequence $$ H^2(X^*,\Z)\stackrel{\alpha}{\ra} E^{0,2}_2\ra E^{3,0}_2=H^3(W^*,\Z).$$ We want to show that $\alpha$ is surjective. If $dimW := w\geq 3$ then $H^3(W^*,\Z)=0$ and we have done. Suppose $w=2$ then $H^3(W^*,\Z)=\Z$; note that the restriction of $-K_X$ gives a non zero class (in fact it is $r+1$ times the generator) in $E^{0,2}_2$ and is mapped to zero in $E^{0,3}_2$ thus the mapping $E^{0,2}_2\ra E^{3,0}_2$ is the zero map and $\alpha$ is surjective. Since $F_0$ is of codimension at least 2 in $X$ the restriction map $H^2(X,\Z)\ra H^2(X^*,\Z)$ is a bijection. By the vanishing of $R_i\pi_*\Ol_X$ we get $H^2(X,\Ol_X)=H^2(W,\Ol_W)=0$ hence also $Pic(X)\ra H^2(X,\Z)$ is surjective. Let $L\in Pic(X)$ be a preimage of a generator of $E^{0,2}_2$. By construction $L_t$ is $\Ol(1)$, for $t\in W^*$. Moreover $(r+1)L=-K_X$ on $X^*$ thus, again by the codimension of $X^*$, this is true on $X$ and $L$ is $\pi$-ample. \end{proof} \section{An approach to the singular case} The following theorem arose during a discussion between us and J.A. Wisniewski; we would like to thank him. The idea to investigate this argument originated with a work of Zhang [Zh2]. For the definition of log-terminal singularity we refer to \cite{KMM}. \begin{Theorem} Let $X$ be an n-dimensional log-terminal projective variety and $E$ be an ample vector bundle of rank $r$. Assume that $K_X+det E$ is nef and let $\pi:X\ra W$ be the contraction supported by $K_X+det E$. Assume also that for any fiber $F$ of $\pi$ $dimF\leq r-1$, that $r\geq (n+1)/2$ and $codim Sing(X)>dim W$. Then $X$ and $W$ are smooth and for any fiber $F\simeq \Proj^{r-1}$. \end{Theorem} \begin{proof} We will prove that $X$ is smooth. Then we can apply proposition (\ref{relscroll}). We consider in this case the associated projective space bundle $Y$ and the commutative diagram \begin{equation} \matrix{{\bf P}(E)=Y&\mapright\varphi&V\cr \mapdown{p}&&\mapdown\psi\cr X&\mapright\pi&W} \end{equation} as in (\ref{dia1}); it is immediate that $Y$ is Gorenstein and log-terminal; in particular it has Cohen-Macaulay singularities. Moreover, as in (3.1) $dim F(\f)\leq dim F(\pi)$ and the map $\f$ is supported by $K_Y+rH$, where $H =\taut + A$, with $\taut$ the tautological line bundle and $A$ a pull back of a ample line bundle from $V$. It is known that a contraction supported by $K_Y+rH$ on a log terminal variety has to have fibers of dimension $\geq (r-1)$ and of dimension $\geq r$ in the birational case (\cite [remark 3.1.2]{AW}). Thus $\f$ is not birational and all fibers have dimension $r-1$; moreover, by the Kobayashi-Ochiai criterion the general fiber is $F\iso \Proj^n$. Imitating the proof of \cite[Prop 1.4]{BS} we have only to show that there are no fibers of $\f$ entirely contained in $Sing(Y)$. Note that, by construction, $Sing(Y)\subset p^{-1}(Sing X)$. Hence no fibers $F$ of $\f$ can be contained in $Sing(Y)$ and therefore the same proof of \cite[Prop 1.4]{BS} applies. It follows that $V$ is nonsingular and $\f:Y\ra V$ is a classical scroll. In particular $Y$ is nonsingular and therefore also $X$ is nonsingular. \end{proof} As a Corollary we obtain Mukai Conjecture 1 in the log terminal case, see also \cite{Zh2}. \begin{Corollary} Let $X$ be an n-dimensional log-terminal projective variety and $E$ an ample vector bundle of rank $n+1$, such that $c_1(E)=c_1(X)$. Then $(X,E)=(\Proj^n,\oplus^{n+1}\Ol_{\Proj^n}(1))$. \end{Corollary} \section{Main theorem} This section is devoted to the proof of the following theorem. \begin{Theorem} Let $X$ be a smooth projective variety over the complex field of dimension $n \geq 3$ and $E$ an ample vector bundle on $X$ of rank $r= n-2$. Then we have \begin{itemize} \item[1)] $K_X + det(E)$ is nef unless $(X,E)$ is one of the following: \begin{itemize} \item[{i})] there exist a smooth $n$-fold, $W$, and a morphism $\phi : X \ra W$ expressing $X$ as a blow up of a finite set $B$ of points and an ample vector bundle $E'$ on $W$ such that $E = \phi^*E'\otimes[-\phi^{-1}(B)]$. \par\noindent Assume from now on that $(X,E)$ is not as in (i) above (that is eventually consider the new pair $(W,E')$ coming from (i)). \item[{ii})] $X = \Proj^n$ and $E =\oplus^{n-2}\Ol(1)$ or $\oplus^{2}\Ol(2)\oplus^{n-4}\Ol(1)$ or $\Ol(2)\oplus^{n-3}\Ol(1)$ or $\Ol(3)\oplus^{n-3}\Ol(1)$. \item[{iii})] $X = \Q^n$ and $E =\oplus^{n-2}\Ol(1)$ or $\Ol(2)\oplus^{n-3}\Ol(1)$ or ${\bf E}(2)$ with ${\bf E}$ a spinor bundle on $\Q^n$. \item[{iv})] $X = \Proj^2 \times \Proj^2$ and $E = \oplus^2\Ol(1,1)$ \item[{v})] $X$ is a del Pezzo manifold with $b_2 = 1$, i.e. $Pic(X)$ is generated by an ample line bundle $\Ol(1)$ such that $\Ol(n-1) = \Ol(-K_X)$ and $E = \oplus^{n-1}\Ol(1)$. \item[{vi})] $X$ is a classical scroll or a quadric bundle over a smooth curve $Y$. \par \item[{vii})] $X$ is a classical scroll over a smooth surface $Y$. \end{itemize} \item[2)] If $K_X + det(E)$ is nef then it is big unless there exists a morphism $\phi : X \ra W$ onto a normal variety $W$ supported by (a large multiple of) $K_X + det(E)$ and $dim(W) \leq 3$; let $F$ be a general fiber of $\phi$ and $E^{\prime}=E_{|F}$. We have the following according to $s = dim W$: \begin{itemize} \item[{i})] If $s = 0$ then $X$ is a Fano manifold and $K_X + det(E) = 0$. If $n\geq 6$ then $b_2(X) = 1$ except if $X=\Proj^3\times\Proj^3$ and $E=\oplus^4\Ol(1,1)$. \item[{ii})] If $s = 1$ then $W$ is a smooth curve and $\phi$ is a flat (equidimensional) map. Then $(F,E')$ is one of the pair described in \cite{PSW}, in particular $F$ is either $\Proj^n$ or a quadric or a del Pezzo variety. If $n \geq 6$ then $\pi$ is an elementary contraction. If the general fiber is $\Proj^{n-1}$ then $X$ is a classical scroll while if the general fiber is $\Q^{n-1}$ then $X$ is a quadric bundle. \item[{iii})] If $s = 2$ and $n \geq 5$ then $W$ is a smooth surface, $\phi$ is a flat map and $(F,E^{\prime})$ is one of the pair described in the Main Theorem of \cite{Fu2}. If the general fiber is $\Proj^{n-2}$ all the fibers are $\Proj^{n-2}$. \item[{iv})] If $s = 3$ and $n \geq 5$ then $W$ is 3-fold with at most isolated singularities and $X$ has at most isolated fibers of dimension n-2; all fibers over smooth point are isomorphic to $\Proj^{n-3}$. \end{itemize} \end{itemize} \item[3)] Assume finally that $K_X + det(E)$ is nef and big but not ample. Then a high multiple of $K_X + det(E)$ defines a birational map, $\f :X \ra X'$, which contracts an "extremal face" (see section 2). Let $R_i$, for $i$ in a finite set of index, the extremal rays spanning this face; call $\rho_i: X \ra W$ the contraction associated to one of the $R_i$. Then we have that each $\rho_i$ is birational and divisorial; if $D$ is one of the exceptional divisors (we drop the index) and $B = \rho (D)$ we have that $dim(B) \leq 1$ and the following possibilities occur: \begin{itemize} \item[{i})] $dimB = 0$, $D = \Proj^{n-1}$ and $D_{|D} = \Ol(-2)$ moreover $E_{|D}\simeq \oplus^{n-2}\Ol(1)$. \item[{ii})] $dimB = 0$, $D$ is a (possible singular) quadric, $\Q^{n-1}$, and $D_{|D} = \Ol(-1)$; moreover $E_{|D} =\oplus^{n-2}\Ol(1)$. \item[{iii})] $dimB = 1$, $W$ and $Z$ are smooth projective varieties and $\rho$ is the blow-up of $W$ along $Z$. Moreover $E_{|F} =\oplus^{n-2}\Ol(1)$. \end{itemize} If $n > 3$ then $\f$ is a composition of "disjoint" extremal contractions as in i), ii) or iii). \label{main} \end{Theorem} \begin{proof}[Proof of part 1) of the theorem \ref{main}] Let $(X,E)$ be a generalized polarized variety and assume that $K_X + det(E)$ is not nef. Then there exist on $X$ a finite number of extremal rays, $R_1, \dots , R_s$, such that $(K_X + det(E))^.R_i < 0$ and therefore, by the remark (\ref{mha}), $l(R_i) \geq n-1$. Consider one of this extremal rays, $R = R_i$, and let $\rho : X \ra Y$ be its associated elementary contraction. Then $L := -(K_X+det(E))$ is $\rho$-ample and also the vector bundle $E_1 := E \oplus L$ is $\rho$-ample; moreover $K_X + det(E_1) = \Ol_X$ relative to $\rho$. To proceed we need a relative version of the theorem in \cite{ABW2} which study the positivity of the adjoint bundle in the case of $rank E_1 = n-1$. More precisely we do not assume that $E_1$ is ample but that it is $\rho$-ample (or equivalently a local statement in a neighborhood of the exceptional locus of the extremal ray $R$). For this we notice that the theorem in \cite{ABW2} is true also in the relative case and can be proved verbatim using the relative minimal model theory instead of the absolute (see \cite{KMM}; see also the section 2 of the paper \cite{AW} for a discussion of the local set up). Assume first that $\rho$ is birational, then $K_X + det(E_1)$ is $\rho$-nef and $\rho$-big; note also that, since $l(R_i) \geq n-1$, $\rho$ is divisorial. Therefore we are in the (relative) case C of the theorem in \cite{ABW2} (see also the proposition (\ref{bd}) with $r = n-1$); this implies that $Y$ is smooth and $\rho$ is the blow up of a point in $Y$. Since $l(R_i) \geq n-1$, the exceptional loci of the birational rays are pairwise disjoint by proposition (\ref{birelementare}). This gives theorem \ref{main} (i): the birational extremal rays have disjoint exceptional loci which are divisors isomorphic to $\Proj^{n-1}$ and which contract simultaneously to smooth distinct points on a $n$-fold $W$. The description of $E$ follows trivially (see also \cite{ABW2}). If $\rho$ is not birational then we are in the case B of the theorem in \cite{ABW2}; from this we obtain similarly as above the other cases of the theorem \ref{main}, with some trivial computations needed to recover $E$ from $E_1$. Note that in the case of fibration over a surface, since all fibers are $\Proj^{n-2}$, then $n-1= l(R)> detE\cdot R_i \geq n-2$, thus $l(R)=n-1$ and $detE\cdot R_i=n-2$. Then $-(det E+K_X)$ is a tautological bundle for the fibration and the fibration is a scroll. This part was also independently proved in \cite{Ma} \end{proof} \begin{proof}[Proof of part 2) of the theorem \ref{main}] Let $K_X+detE$ be nef but not big; then it is the supporting divisor of a face $F = (K_X+detE)^{\bot}$. Using (\cite{KMM}) we can say that there exists a map $\pi:X\rightarrow W$ which is given by a high multiple of $K_X+detE$ and which contracts the curves in the face. Since $K_X+detE$ is not big we have that $dimW 3$, proposition (\ref{n=4}). Secondly, by the inequality (\ref{diswis}), we have $$dimE(R)+dimF(\pi)\geq 2n - 3.$$ Therefore $dimE(R)=n-1$ and either $dimF(\pi) = n-1$ or $dimF(\pi) = n-2$, thus $n-1\geq l(R)\geq n-2$. If $B := \rho (E)$ and $D=E(R)$ this implies that either $dimB = 0$ or $1$. If $dimB = 1$ then $dim F(\pi) = n-2$ for all fibers (note that since the contraction $\pi$ is elementary there cannot be fiber of dimension $n-1$); thus we can apply proposition (\ref{bd}) with $r = n-2$. This will give the case 3-(iii) of the theorem. Let now $dimB=0$ and consider again the construction in section (\ref{tech}), in particular we refer to the diagram (\ref{dia1}). Let $S$ be the extremal ray contracted by $\varphi$; note that $l(S)\geq n-2$ and that the inequality (\ref{diswis}) gives $$dimE(S)+dimF(\f)\geq 3n - 6;$$ in particular, since $dim F(\f) \leq dim F(\pi) $, we have two cases, namely $dimE(S) = 2n-5$ and $dimF(\f) = n-1$ or $dimE(S) = 2n-4$ and $dimF(\f) = n-1$ or $n-2$. The case in which $dimE(S) = 2n-5$ will not occur. In fact, after "slicing", (see \ref{adj}), we would obtain a map $\varphi^{\prime}=\varphi_{|Z}$ which would be a small contraction supported by a divisor of the type $K_Z+(n-2)L$ but this is impossible by the classification of \cite[Th 4]{Fu1} (see also \cite{An}). Hence $dimE(S)=2n-4$, that is also $\varphi$ is divisorial and $E(S)\cdot l_p=0$, where $l_p$ is a line in $F(p)$. In particular $E(S)=p^*D$. Suppose that the general fiber of $\varphi$, $F(\f)$, has dimension $n-2$. After slicing we obtain a map ${\varphi}^{\prime}=\varphi_{|Z}: Z \ra T$ supported by $K_Z+(n-2)L$, where $L={\taut}_{|Z}$. This map contracts divisors $\overline{D}$ in $Z$ to curves; by (\cite[Th 4]{Fu1}) we know that every fiber $F$ of this map is $\Proj^{n-2}$ and that $\overline{D}_{|F} = \Ol(-1)$ (actually this map is a blow up of a smooth curve in a smooth variety). In particular there are curves in $Y$, call them $l$, such that $-E(S).l = 1$. We will discuss this case in a while. Assume now that the general fiber and therefore all have dimension $n-1$. \begin{Lemma} In this hypothesis $l(R)=n-2$. \end{Lemma} Let $C$ be a minimal curve in $R$ (see (\ref{mindef})), $\nu:\Proj^1\ra C$ its normalization, $\tilde{\nu}:Y_C=\Proj(\nu^*E_{|C})\ra Y$ the induced morphism and $\xi_C$ the tautological bundle of $Y_C$, note that $\tilde{\nu}^*\taut=\xi_C$. Let $g:Y_C\ra F(\psi)_w$ the morphism induced by $\f$ on $Y_C$ and $$Y_C\stackrel{\alpha}{\ra} V_1\stackrel{\beta}{\ra} F(\psi)_w$$ its Stein factorization. Assume by contradiction that $l(R)=n-1$, then $\nu^*(E_{|C})=\Ol(2)\oplus\Ol(1)^{\oplus n-3}$, hence $Y_C$ has two contractions, the scroll structure and a blow down to $\Proj^{n-2}$. Let $\tilde{l}$ be a line contracted by the blow down then $\tilde{l}$ is contracted by $g$. In fact by projection formula $\taut\cdot \tilde{\nu}_*\tilde{l}=\xi_C\cdot\tilde{l}=1$, thus by the commutativity of the diagram $\tilde{\nu}_*\tilde{l}$ is a minimal curve in S. Since $\alpha$ cannot contract all $Y_C$ then $\alpha$ is the blow down. Since $dimF(\pi)=dimF(\f)$ by hypothesis, then by remark (\ref{fdim}) all fibers $F(\psi)$ have dimension $n-3$. So we get the contradiction that $\beta:\Proj^{n-2}\ra F(\psi)_w$ is a finite map between two varieties of different dimension. Slicing we obtain a map ${\varphi}^{\prime}=\varphi_{|Z}: Z \ra T$ supported by $K_Z+(n-2)L$, where $L={\taut}_{|Z}$. This map contracts divisors $\overline{D}$ in $Z$ to points; by (\cite{Fu1}) we know that these divisors are either $\Proj^{n-1}$ with normal bundle $\Ol(-2)$ or $\Q^{n-1}\subset \Proj^n$ with normal bundle $\Ol(-1)$. In the latter case we have as above that there are curves $l$ in $Y$, such that $-E(S).l = 1$. In these cases observe that $E(S)=p^*(D)$ and $K_X+(n-2)(-D)$ is a supporting divisor for $\pi$. Then by \cite{Fu1} we conclude that $(D,D_{|D})$ is one of the pair listed in the theorem and the theory of uniform bundle allows to recover easily $E_{|D}$ (\cite{OSS}). It remains the case in which ${\varphi}^{\prime}=\varphi_{|Z}: Z \ra T$ contracts divisors $\overline{D}= \Proj^{n-1}$ with normal bundle $\Ol(-2)$ to points. We can apply proposition (\ref{fujita}) and show that the singularities of $W$ are the same as those of $T$. Then, as in (\cite{Mo1}), this means that we can factorize $\pi$ with the blow up of the singular point. Let $X^{\prime}=Bl_{w}(W)$, then we have a birational map $g:X\rightarrow X^{\prime}$. Note that $X^{\prime}$ is smooth and that $g$ is finite. Actually it is an isomorphism outside $D$ and cannot contract any curve of $D$. Assume to the contrary that $g$ contracts a curve $C^{\prime} \subset D$; let $N\in Pic(X^{\prime})$ be an ample divisor then we have $g^*N\cdot C^{\prime}=0$ while $g^*N\cdot C\not=0$ contradiction. Thus by Zarisky's main theorem $g$ is an isomorphism. This gives the case in 3)i). \end{proof} \begin{thebibliography}{999999} \bibitem[An]{An} Andreatta,M., \textit{Contractions of Gorenstein polarized varieties with high nef value}, Math. 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