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\begin{document} 
 
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   UNIVERSITY OF  KAISERSLAUTERN \hfill  Summerterm 1998 \\
   Mathematical Department \hfill 5th July, 1998 \\
   Gert-Martin Greuel \hfill Thomas Keilen\\
   \vspace{0.5cm}

   \begin{center}
     \LARGE \bf Algebraic Geometry \\[0.1cm] \normalsize Exercise set 12
   \end{center}
   \vspace{0.5cm}
   \normalsize
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   \textbf{Due date:} 13th July, 1998\hfill \textbf{10\ a.\ m.}
   \\[0.2cm]
   
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   \setcounter{exercise}{34}

   \begin{exercise}
     Let $X$ be a projective variety over the algebraically closed
     field $K$, $L$ an invertible sheaf on 
     $X$, and let $L$ be globally generated by $s_0,\dots,s_n\in
     \Gamma(X,L)$ (i.\ e.\ $s_{0x},\dots,s_{nx}$ generate $L_x$ 
     for every $x\in X$). Denote by $\varphi:X\rightarrow\P_K^n$ the
     morphism defined by $s_0,\dots,s_n$.
     \begin{enumerate}
        \item If the vector subspace $V\leq \Gamma(X,L)$ spanned by
          $s_0,\dots,s_n$ has dimension $r$, then $\varphi(X)$ is
          contained in a linear subspace of $\P_K^n$ of dimension $r$.
        \item If $t_0,\dots,t_m$ ($m\geq n$) is another system of generators of
          $V$ generating at the same time $L$, and if
          $\psi:X\rightarrow\P_K^m$ denotes the corresponding
          morphism, then there exists a linear subspace
          $Y\subseteq\P_K^m$ of dimension $m-n-1$ and an automorphism
          $\alpha$ of $\P_K^n$ such that
          $\varphi=\alpha\circ\pi_Y\circ\psi$, where
          $\pi_Y:\P_K^m\setminus Y\rightarrow\P_K^n$ denotes the
          linear projection with center $Y$.
        \item If $X=\P_K^\mu$, then one of the following two cases comes 
          true:
          \begin{enumerate}
             \item $\im(\varphi)$ is a single point, and
               $L\cong\ko_X$, or
             \item $\dim(\im(\varphi))=\mu\leq n$, and
               $L\cong\ko_X(d)$ for some $d\geq 0$.
          \end{enumerate}
        \item In the second case of c. we have moreover
          $$\varphi=\alpha\circ\pi_Y\circ\rho_d,$$ where
               $\rho_d:\P_K^\mu\rightarrow\P_K^N$ is the $d$-uple
               Veronese embedding, $\pi_Y$ a linear projection along a 
               suitable linear subspace of $\P_K^N$, and $\alpha$ an
               automorphism of $\P_K^n$.\\
               In particular, $\varphi$ is quasi-finite.
     \end{enumerate}
     \small (Hint, since we have a \emph{fully faithful} functor identifying
     the projective variety $\P_K^n$ with the scheme $\P_K^n$, it
     suffices to investigate the above morphisms on the level of
     varieties. Then $\varphi$ is just given by sending a closed point 
     $p\in X$ to the point $(s_0(p):\dots:s_n(p))\in\P_K^n$, where
     $s_i(p)$ denotes the residue class of the germ of $s_i$ at the
     point $p$ in $L_p/\m_{X,p}L_p\cong K$. Thus for a.\ you just have to
     find some linear equations, which the coordinates of $\varphi(p)$ 
     satisfy. For $b.$ note that e.\ g.\ the projection with center
     $Y=V(x_0,\dots,x_n)$ is given by $(q_0:\dots:q_m)\mapsto
     (q_0:\dots:q_n)$. Then basically all you have to know is, that
     given two systems with $n$ vectors generating the same subspace
     of dimension $r$ of an $n$-dimensional vectorspace, there is a
     linear automorphism of the vectorspace transforming one into the
     other.)
   \end{exercise} 

   \begin{exercise-remark}
     The above statements remain true if we require $X$ only to be any
     scheme over a (not necessarily algebraically closed) field
     $K$. But then we can no longer interpret the morphisms in the  
     above way.
   \end{exercise-remark}

   \begin{exercise}
     Prove proposition 5.7 of the lecture, i.~e.~show that if $A$ is a
     noetherian domain, then the following are equivalent:
     \begin{enumerate}
        \item $A$ is a unique factorization domain.
        \item $X=\Spec(A)$ is normal, and $\Cl(X)=0$.
     \end{enumerate}
   \end{exercise}

   \begin{exercise-break}
     \begin{enumerate}
        \item Show that if $C$ is a projective plane curve of degree
          $d$, then $\Cl(\P_K^2\setminus C)=\Z/d\Z$.
        \item Let $K$ be any field, $A=K[X,Y,Z]/(XY-Z^2)$, and
          $X=\Spec(A)$. Show that $\Cl(X)=\Z/2\Z$.
     \end{enumerate}
   \end{exercise-break}

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