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   \Blatt{5}
   \Abgabe{03/12/2001}


   \Nr
   Let $K$ be any field. Show that the map $\varphi:\A_K^1\rightarrow\A_K^2:t\mapsto
   \big(t^2,t^3\big)$ induces a homeomorphism from $\A_K^1$ to the cuspidal
   cubic $V\big(x^3-y^2\big)$ which is a morphism, but not an isomorphism.

   \Nr
   For the following morphisms $f$ calculate with the aid of \textsc{Singular}
   the vanishing ideal $I\Big(\overline{\im(f)}\Big)$ and decide
   whether $\im(f)$ is closed or not.
   \begin{enumerate}
   \item $f:\A_\C^1\rightarrow \A_\C^3\;:\;t\mapsto
     \big(t^3,t^4,t^5\big)$.
   \item $f:\A_\C^2\rightarrow \A_\C^3\;:\;(s,t)\mapsto
     \big(s,st,t^2\big)$. 
   \item $f:\A_\R^2\rightarrow \A_\R^3\;:\;(s,t)\mapsto
     \big(s,st,t^2\big)$, this is slightly more difficult. 
   \end{enumerate}
   {\scriptsize Hint: The command \texttt{eliminate} can be used in
     order to eliminate variables from an ideal.}

   \Nr
   \begin{enumerate}
   \item Let $K\subset L$ be a field extension, where $L$ is algebraically
     closed.  Let $f=\sum_{|\alpha|=0}^na_\alpha
     \underline{x}^\alpha\in K[\underline{x}]$ be a polynomial and
     consider the ideal
     $$I=\left( \sum_{\beta+\gamma=\alpha} b_{\beta}\cdot
       c_{\gamma}-a_\alpha\; \bigg|\; |\alpha|=0,\ldots,2n-2\right) \lhd
     L\big[b_{\beta},c_{\gamma}\;\big|\; 
     |\beta|,|\gamma|=0,\ldots,n-1\big],$$
     where $a_\alpha=0$ for $|\alpha|=n+1,\ldots,2n-2$.

     Show that $f$ is irreducible over $\overline{K}$ if and only if
     $I=L\big[b_{\beta},c_{\gamma}\;\big|\; 
     |\beta|,|\gamma|=0,\ldots,n-1\big]$.
   
     {\scriptsize Remark: This gives a criterion to check, whether a
       hypersurfes given by a polynomial $f\in\Q[\underline{x}]$ is
       irreducible over $\C$.}
   \item Check, using this criterion and \textsc{Singular}, whether the polynomials
     $f=x^4+4x^3y+2x^2y^2-4xy^3+y^4+2x^2+4xy+1$ and 
     $g=x^4+4x^3y+2x^2y^2-4xy^3+y^4+2x^2+4xy-1$ 
%     $g=z^6+2y^2z^3+y^4+2xz^3+2xy^2+x^2+2$ 
     are irreducible over $\Q$ and / or $\C$. 
   \end{enumerate}
   {\scriptsize Hint: The command \texttt{reduce(1,std(I));} returns
     zero if and only if the ideal $I$ is the whole ring.}

   \Nr
   Let $K$ be any 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,
   if $f^*:K[W]\rightarrow K[V]$ is a finite ring-homomorphism, then $f$ is quasi-finite,
   i.~e.~all fibres of $f$ are finite.

   {\scriptsize Hint: Show first that if $k$ is a field and $k\subseteq R$ is a ring
     extension, such that $R$ is a
     finite-dimensional $k$-vectorspace, then $R$ can have only
     finitely many maximal ideals. 
%     Finally, you may consider the ring
%     $K[V]/f^*(\m_p) K[V]$ for $p\in W$.
     }

   {\scriptsize Remark: Knowing a bit about localisations one may more
     generally show that if $f^*:A\rightarrow B$ is a finite
     ring-homomorphism, then the induced map $f:\Spec(B)\rightarrow
     \Spec(A)\;:\; P\mapsto \varphi^{-1}(P)$ is quasi-finite.}


   
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