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   \large UNIVERSITY OF  KAISERSLAUTERN \hfill  Winterterm 1997/98 \\
   Mathematical Department \hfill 28th November, 1997 \\
   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:} 8th December 1997
   \\[0.2cm]
   
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   \setcounter{exercise}{19}

   \begin{exercise}
     Let $K$ be an algebraically closed field.
     \begin{enumerate}
        \item Show, if an affine variety $V$ is isomorphic to a projective
          variety $W$, then both consist of an equal finite number of
          points.
        \item Let $H\subset \P^n_K$ be a hypersurface and
          $V\subset\P^n_K$ be a subvariety with $\dim(V)\geq 1$, then
          $V\cap H\not= \emptyset$. \\
          \small (Hint, use Exercise 18 and part a.)
     \end{enumerate}
   \end{exercise}

   \begin{exercise}
     Let $K$ be algebraically closed. \\
     Show that $V:=\A^2_K\backslash\menge{(0,0)}$ is not affine, i.\ e.\
     is not isomorphic to an affine variety.\\
     \small (Hint, show that the inclusion $i:V\hookrightarrow \A_K^n$ 
     induces an isomorphism $i^*:K[x,y]\rightarrow \ko(V)$.)
   \end{exercise}

   \begin{exercise}[Central Projection]
     Let $K$ be any field, $\P_K^{n-1}\cong H\subset\P_K^n$ be a hyperplane and
     $P\in\P_K^n$ with $P\not\in H$. We define the \emph{central
       projection} through $P$ onto $H$,
     $\varphi:\P_K^n\backslash\menge{P}\mapsto \P^{n-1}_K$, by mapping 
     a point $Q$ to the unique intersection point of the line
     $\overline{PQ}$ with $H$. 
     Show that $\varphi$ is a morphism. \\
     \small (Hint, use a linear change of coordinates to reduce to the 
     situation $H=\menge{x_0=0}$ and $P=(1:0:\ldots:0)$.)
   \end{exercise}

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