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   \Blatt{8}
   \Abgabe{07/01/2002}

   For simplicity we assume that $K$ is an algebraically closed field.

   \Nr
   Let $V\subseteq \A_K^n$ be a quasi-affine variety. Show, show that
   $f:V\rightarrow K$ is a regular function if and only if for any
   $p\in V$ there there are polynomials $g,h\in K[x_1,\ldots,x_n]$ and
   an open neighbourhood $p\in U(p)\subseteq V$  such that $h(q)\not=
   0$ and $f(q)=\frac{g(q)}{h(q)}$ for any $q\in U(p)$. 

   \Nr
   Let $V$ be a quasi-projective variety and $f\in\ko(V)$ be a regular
   function. Show that $f$ is continous.

   \Nr
   \begin{enumerate}
   \item Let $H=V(f)$ be a hypersurface in $\P_K^n$ and $V\subseteq
     \P_K^n$ be any projective variety.  Show that $V\setminus H$ is
     isomorphic to an affine algebraic set.
     
     {\scriptsize Hint: First note that it suffices to consider the
       case $V=\P_K^n$, and then prove the case that $\deg(f)=1$.
       Finally, if $f$ has degree $d$, use the Veronese embedding
       $\rho_d$.}
   \item Let $H=V(f)$ be a hypersurface in $\A_K^n$ and $V\subseteq
     \A_K^n$ be any affine variety. Show that $V\setminus H$ is
     isomorphic to an affine algebraic set.
   \end{enumerate}   

   \Nr\textbf{[Rational Functions]}\\[0.1cm]
   $Q$ be the quadric $V(xw-yz)$
   in $\P_K^3$, and $f\in R(Q)$ be the rational function on $Q$ induced by
   $(w:x:y:z)\mapsto\frac{x}{y}$.
   \begin{enumerate}
   \item Given $\emptyset\not=U\subseteq Q$ open and $g,h\in
     K[w,x,y,z]_d$ such that $f\equiv \frac{g}{h}$ on $U$. Show,
     that then $$yg-xh\equiv 0 \mbox{ on } Q,$$
     and deduce, that there exist polynomials $h_1,h_2\in
     K[w,x,y,z]_{d-1}$ such that $$h=yh_1+wh_2.$$
   \item Show, the pol set of $f$ is $\{(w:x:y:z)\in Q|
       w=y=0\}$.
   \item Show, there exist no polynomials $g,h\in K[w,x,y,z]$
     such that $f\equiv \frac{g}{h}$ on the whole definition set
     of $f$.
   \end{enumerate}
   {\scriptsize Hint: For a. look at the points $P_{x,z}=(0:x:0:z)$, for
   c. at $P_{w,z}=(w:0:0:z)$ and at $P_{w,y}=(w:0:y:0)$. Moreover,
   use in a. that $Q$ is 
   irreducible!}
%   (Remark: $Q$ can be identified with the Segre embedding of
%   $\P_1\times\P_1$ in $\P_3$.)
   

   
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