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   \large UNIVERSITY OF  KAISERSLAUTERN \hfill  Winterterm 1997/98 \\
   Mathematical Department \hfill 02nd December, 1997 \\
   Gert-Martin Greuel \hfill Thomas Keilen\\
   \vspace{0.5cm}

   \begin{center}
     \LARGE \bf Introduction to Algebraic Geometry \\
   \end{center}
   \vspace{0.5cm}
   \normalsize
   \parindent0cm
   \textbf{Due date:} 15th December, 1997
   \\[0.2cm]
   
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   \setcounter{exercise}{22}

   \begin{exercise}
     Let $K$ be any field, $V$ and $W$ be varieties over $K$ and
     $\varphi:V\rightarrow W$ be any map. If $\menge{ V_i|i\in I}$ is
     an open covering of $V$ and $\menge{ W_i|i\in I}$ is
     an open covering of $W$ such that $\varphi(V_i) \subseteq W_i$
     for all $i\in I$.\\
     Then the following are equivalent:
     \begin{enumerate}
        \item $\varphi$ is a morphism.
        \item $\varphi_{|}:V_i\rightarrow W_i$ is a morphism for all
          $i\in I$.
     \end{enumerate}
     If moreover $W_i\subseteq \A_K^{n_i}$ is quasiaffine, these are equivalent
     to: 
     \begin{enumerate}\setcounter{enumi}{2}
        \item $\varphi^*\ko(W_i)\subseteq\ko(V_i)$ for all $i\in I$,
          where $\varphi^*(h)=h\circ \varphi$ for $h\in\ko(W_i)$.
        \item For all $i\in I$ we have
          $\varphi^*(x^{(i)}_j)$ is regular for all $j=1,\ldots,n_i$
          (where $x_j^{(i)}$ is the j-th coordinate function of $\A^{n_i}_K$).
     \end{enumerate}
   \end{exercise}

   \begin{exercise}
     Show that the map $\varphi:\A_K^1\rightarrow\A_K^2:t\mapsto
     (t^2,t^3)$ induces a homeomorphism from $\A_K^1$ to the cuspidal
     cubic $V(x^3-y^2)$ which is a morphism, but not an isomorphism.
   \end{exercise}
   \begin{exercise}[Segre embedding]
     Let $K$ be any field, and let $\P_K^r\times\P_K^s$ be a product
     of projective spaces, i.\ e.\ equipped with its Zariski
     topology. We define the Segre embedding
     $\varphi:\P_K^r\times\P_K^s\rightarrow\P_K^N$ with $N=rs+r+s$ by
     $$(p_0:\ldots:p_r)\times (q_0:\ldots:q_s)\mapsto
     (p_0q_0:\ldots:p_0q_s:p_1q_0:\ldots
     p_1q_s:\ldots:p_rq_0:\ldots:p_rq_s).$$ 
     \begin{enumerate}
        \item Show, that for
          $\vartheta:K[z_{ij}|i=0,\ldots,r,j=0,\ldots,s]\rightarrow
          K[x_0,\ldots,x_r,y_0,\ldots,y_s]:z_{ij}\mapsto x_iy_j$ we
          have $\im(\varphi)=V(\ker(\vartheta))$.\\
          {\small (Hint, if
          $R=(R_{00},\ldots,R_{rs})\in V(\ker(\vartheta))$ with
          $R_{kl}=1$, then $R=\varphi(P\times Q)$ with
          $P=(R_{0l},\ldots,R_{rl})$ and
          $Q=(R_{k0},\ldots,R_{ks})$.)}
        \item Show, $\varphi$ is a morphism.\\
          {\small (Hint, cover $\P_K^N$ by the affine charts
          $W_{ij}=\menge{z_{ij}\not=0}$ and use exercise 23. Note, the 
          affine coordinate functions on $W_{ij}$ are
          $\frac{x_{kl}}{x_{ij}}$, $(k,l)\not=(i,j)$.)}
        \item Show, that $\varphi$ is injective.
        \item Show,
          $\varphi^{-1}:\im(\varphi)\rightarrow\P_K^r\times\P_K^s$ is
          a morphism.\\
          {\small (Hint, use the fact, that
          $V_{ij}=\varphi^{-1}(W_{ij})$ is affine with affine
          coordinate functions $\frac{x_k}{x_i}$, $k\not=i$, and
          $\frac{y_l}{y_j}$, $l\not=j$.)}
     \end{enumerate}
   \end{exercise}

\end{document} 
