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   \renewcommand{\ABGABETAG}{\textbf{Friday}}
   \Abgabe{\textbf{23/11/2001 !!!}}

   \Nr 
   Let $K$ be any field, $V=V(xz,yz)\subseteq \A_K^3$ and
   $p=(0,0,1),q=(1,0,0),0=(0,0,0)\in V$. Find $\dim(V,p)$, $\dim(V,q)$
   and $\dim(V,0)$.

   \Nr
   Let $K$ be an algebraically closed field, $V\subseteq \A_K^n$ an
   irreducible algebraic variety of dimension $r$ and $H\subset\A_K^n$ a
   hypersurface with $V\not\subset H$. Show, that any irreducible
   component of $V\cap H$ has dimension $r-1$. In particular, if $V\cap
   H\not=\emptyset$, then $\dim(V\cap H)=r-1$.

   \Nr
   Let $K$ be any field, $I=\big(y^2z-xz^2,x^3yz-x^2y^3-xz^3+y^2z^2\big)\unlhd
   K[x,y,z]$. Find the decomposition of $V(I)$ into irreducible
   components. 

   {\scriptsize Hint: You may use the procedures \texttt{primdecGTZ}
     or  \texttt{minAssGTZ}
     in the library \texttt{primdec.lib} of \textsc{Singular} to get an idea!}

   \Nr
   Let $K$ be an algebraically closed field, let $V\subseteq \A_K^n$
   and $W\subseteq \A_K^m$ be two affine 
   algebraic varieties and let $f:V\rightarrow W$ be a morphism. Show:
   \begin{enumerate}
   \item $f(p)=q$ if and only if $f^{*-1}(\m_p)=\m_q$, where $p\in V$
     and $q\in W$.
   \item $f^*$ is injective if and only if $f$ is dominant.
   \item $f^*$ is surjective if and only if $f(V)=V\big(\Ker(f^*)\big)$ is closed in $W$
     and $f_|:V\rightarrow f(V)$ is an isomorphism.
   \end{enumerate}
   {\scriptsize Remark: The assumption $K$ algebraically closed is in
     a certain sense not
     necessary. One can more generally show, if $f^*:A\rightarrow B$ is any ring-homomorphism,
     then b.~and c.~hold also for the induced map $f:\Spec(B)\rightarrow 
     \Spec(A)\;:\; P\mapsto \varphi^{-1}(P)$. } 

   
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