
\documentclass[12pt]{article}

\usepackage[all]{xypic}
%\usepackage{latexsym}
%\usepackage{amssymb}
%\usepackage{theorem}
%\usepackage{euler}
%\usepackage{epic}
%\usepackage{eepic}
%\usepackage{color}
%\usepackage{curves}

\input msc.sty



\setlength{\topmargin}{30mm}     % Grundlinie der 1. Zeile 30mm von Oberkante
\addtolength{\topmargin}{-1in}
\addtolength{\topmargin}{-\headsep}
\addtolength{\topmargin}{-\headheight}
\addtolength{\topmargin}{-\topskip}

\setlength{\textheight}{267mm} % Unterkante der Fussbox 30pt-voffset ueber Rand
\addtolength{\textheight}{\topskip}
\addtolength{\textheight}{-\footskip}
\addtolength{\textheight}{-30pt}

\setlength{\oddsidemargin}{-1in} % Linker Rand 30mm vom Papierrand
\addtolength{\oddsidemargin}{30mm}
\setlength{\evensidemargin}{\oddsidemargin}

\setlength{\textwidth}{150mm} % Rechter Rand 30mm vom Papierrand




\begin{document}

   \pagestyle{empty}
   \parindent0cm
   UNIVERSITY OF  KAISERSLAUTERN \hfill  Summerterm 1998 \\
   Mathematical Department \hfill 25th June, 1998 \\
   Gert-Martin Greuel \hfill  Christoph Lossen\\
   \vspace{0.5cm}

   \begin{center}
     \LARGE \bf Algebraic Geometry \\[0.1cm] \normalsize Exercise set 11
   \end{center}
   \vspace{0.5cm}
   \normalsize
   \parindent0cm
   \textbf{Due date:} 06th July, 1998 \hfill \textbf{10 a.\ m.}
  
   \parindent0cm

   \bigskip
   \setcounter{exercise}{31}


   \begin{exercise} %(vector bundles)
     A morphism of schemes \mbox{$E
       \stackrel{\pi}{\to} 
       X$} is called a {\em vector bundle of rank $n$ over 
       $X$,\/} 
     if there exists an open covering
     \mbox{$\mathfrak{U}=(U_i)_{i\in I}$} of $X$ and isomorphisms
     \mbox{$\varphi=(\varphi_i)_{i\in I}$} over $U_i$, 
     $$ \xymatrix@C=6pt@R=10pt{ \pi^{-1}(U_i)  \ar[dr]_{\pi} 
       \ar[rr]^{\varphi_i}_{\cong} & & 
       \A^n_{U_i} \ar[dl]  \\
       & U_i}
     $$
     such that for any $i,j$, and for any open affine subset \mbox{$\Spec A \subset
       U_i\cap U_j$} the automorphism \mbox{$\varphi_j \circ \varphi_i^{-1} =:\psi$}
     is given by a {\em linear automorphism\/} $\theta$ of
     \mbox{$A[X_1,\dots,X_n]$}.
     \footnote{Recall, that for any scheme $Y$ we defined
       \mbox{$\A^n_{Y}:=\Spec(\Z[X_1,\ldots,X_n])\times_{\Spec(\Z)} Y$}, 
       and that for $Y=\Spec(A)$ we 
       have $\A^n_Y\cong \A^n_A=\Spec(A[X_1,\dots,X_n])$.}
     We call the data $(\mathfrak{U},\varphi)$ a vector
     bundle atlas for $\pi$. 
     
     A morphism of rank $n$ vector bundles \mbox{$g:(E \stackrel{\pi}{\to} X) \lra
       (E' \stackrel{\pi'}{\to} X)$} 
     is a morphism of schemes such that the following diagram commutes
     $$ \xymatrix@C=6pt@R=10pt{ E  \ar[dr]_{\pi} 
       \ar[rr]^{g}_{\cong} & & E' \ar[dl]^{\pi'}  \\
       & X}
     $$
     and such that there exist vector bundle atlasses $(\mathfrak{U}, \varphi)$
     and $(\mathfrak{U}', \varphi')$ for $\pi$, respectively $\pi'$, such that the 
     covering \mbox{$ \mathfrak{U}\cup \mathfrak{U}'$} together with the
     isomorphisms \mbox{$\varphi_i, \,\varphi'_i \circ g$} defines also a vector
     bundle atlas for $\pi'$.
     
     A (local) {\em section\/} of the rank $n$ vector bundle
     \mbox{$E \stackrel{\pi}{\to} X $} is 
     a morphism \mbox{$\sigma:U\to E$}, \mbox{$U\subset X$} open, such that
     \mbox{$\pi \circ \sigma = \id_U$}. 
     
     \medskip
     To any rank $n$ vector bundle \mbox{$E \stackrel{\pi}{\to} X$} 
     we can associate
     its {\em sheaf of local sections,}\/ 
     denoted by \mbox{$\ks_{E/X} $}. Prove that
     \begin{enumerate}
     \item[(a)] $\ks_{E/X}$ is a locally free $\ko_X$-module of rank $n$.
     \item[(b)] If \mbox{$(E \stackrel{\pi}{\to} X) \cong (E' \stackrel{\pi'}{\to}
         X)$}, then \mbox{$\ks_{E/X} \cong \ks_{E'/X}$}.
     \end{enumerate}
     
     \medskip \noindent
     (Indeed, this association creates an equivalence between the
     category of isomorphismclasses of vector
     bundles over $X$  and the category of isomorphismclasses of
     locally free sheaves on $X$.)  
   \end{exercise}
   
   
   
   \medskip
   \newpage
   
   \begin{exercise}
     \begin{itemize}
     \item[(a)]
       Let $\kl$ be an invertible sheaf on a noetherian scheme $X$. Prove that the
       following are equivalent:
       \begin{enumerate}
       \item[(1)] $\kl$ is ample.
       \item[(2)] $\kl^m$ is ample for all \mbox{$m\geq 1$}.
       \item[(3)] $\kl^m$ is ample for some \mbox{$m\geq 1$}.
       \end{enumerate}
       Moreover, show that if $X$ is a scheme over $\Spec(A)$, for $A$
       noetherian, and if \mbox{$\kl^m$} is very ample (over
       $\Spec(A)$)  for some \mbox{$m\geq 1$},
       then $\kl$ is ample.
     \item[(b)] Let $A$ be a noetherian ring.  For which \mbox{$d\in
         \Z$} are the (invertible) sheaves 
       \mbox{$\ko_{\P^n_A}(d)$} ample? 
     \end{itemize}
   \end{exercise}
   
   \medskip
   
   \begin{exercise}
     Let \mbox{$n\neq m$}. Prove that \mbox{$\ko_{\P^r_K}(n) \not\cong
       \ko_{\P^r_K}(m)$}.  
   \end{exercise}
   



\end{document}


