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   \Blatt{9}
   \Abgabe{14/01/2002}

   {\small Remember: A variety is called \emph{affine}, if it is isomorphic to
   a closed subvariety of some $\A^n_K$.}

   \Nr\textbf{[Two Applications of Exercise 29]}\\[0.1cm]
   Let $K$ be an algebraically closed field.
   \begin{enumerate}
   \item If $V$ is any quasi-projective variety and $p\in V$, then
     $p$ has an affine open neigbourhood $U(p)\subseteq V$.
   \item If $H\subset\P_K^n$ is a hypersurface and $V\subseteq\P_K^n$
     is a projective subvariety of dimension $\dim(V)\geq 1$, then
     $H\cap V\not= \emptyset$.

     {\scriptsize Hint, recall that only finitely many points form varieties which
       are affine \emph{and} projective.}
   \end{enumerate}


   \Nr
   Find affine varieties $V$ and $W$ such that $K[V]\not\cong K[W]$,
   but $R(V)\cong R(W)$.


   \Nr
   Let $K$ be an algebraically closed field. Show that
   $\A_K^2\setminus\{0\}$ is not affine.

   {\scriptsize Hint: Show that the inclusion
     $i:\A_K^2\setminus\{0\}\hookrightarrow \A_K^2$ induces an
     isomorphism
     $i^*:K[x,y]\rightarrow\ko\big(\A_K^2\setminus\{0\}\big)$. For the
     surjectivity show that every $f\in\ko(V)$ has the form
     $f=\tfrac{g}{h}$ and consider $\dim\big(V(h)\big)$.}

   
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