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   \large UNIVERSITY OF  KAISERSLAUTERN \hfill  Winterterm 1997/98 \\
   Mathematical Department \hfill 23rd January, 1998 \\
   Gert-Martin Greuel \hfill Thomas Keilen\\
   \vspace{0.5cm}

   \begin{center}
     \LARGE \bf Introduction to Algebraic Geometry \\
   \end{center}
   \vspace{0.5cm}
   \normalsize
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   \textbf{Due date:} 02nd February, 1998 
   \\[0.2cm]
   
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   \setcounter{exercise}{40}

   \begin{exercise}
     Let $X$ and $Y$ be varieties over $K$, $K$ algebraically closed. 
     Show the following:
     \begin{enumerate}
        \item If $X$ is complete, then every closed subvariety 
          of $X$ is complete.
        \item If $f:X\rightarrow Y$ is a morphism with $X$ complete
          and $Y$ separated,  then
          $(f(X),\ko_{f(X)})$ is a closed subvariety of $Y$ and it is complete.
        \item Find an example for a morphism of (affine) varieties, where the
          image is not a \emph{sub}variety, that is the image is not
          locally closed.
        \item If $X$ is quasi affine and complete, then $X$ consists of finitely
          many points.
        \item If $f:X\rightarrow Y$ is a morphism, where
          $X$ is complete and
          $Y$ is quasi affine, then $f(X)$ consists of finitely many points.
     \end{enumerate}
     {\small (Hint, in a. use without proving that, given subvarieties
       $X'\subseteq X$ and $Y'\subseteq Y$, on $X'\times Y'$ the product
       structure and the subspace structure induced by $X\times Y$
       coincide. For c. think of constructible sets.)}
   \end{exercise}

   \begin{exercise}
     Let $K$ be algebraically closed.
     Show that $\P_K^1\times\P_K^1$ is birationally equivalent to
     $\P_K^2$, in particular $\P_K^1\times\P_K^1$ is rational.\\
     {\small (You could construct the birational map from $\P_K^2$ to
       $\P_K^1\times \P_K^1$ explicitly by using two central projections of $\P_K^2$
       with different centres. This, by the way, coincides with
       blowing up $\P_K^2$ in these two points and then blowing down
       the line connecting these two points! But there is a much
       simpler abstract argument, that needs only half a line.)}
   \end{exercise}

   \begin{exercise}
     Let $K$ be algebraically closed. Let $\sigma:S\rightarrow\A_K^2$
     be the blow up of $\A_K^2$ in $0$. Denote by $C$ the cuspidal cubic 
     $\menge{Y^2-X^3=0}\subset \A_K^2$, and by $\tilde{C}$ the strict
     transform of $C$ in $S$. \\
     Show that $\tilde{C}$ is isomorphic to $\A_K^1$ and deduce, that
     $\sigma_|:\tilde{C}\rightarrow C$ is not an isomorphism, although 
     it is a homeomorphism.
   \end{exercise}

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