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\begin{document} 

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   \large UNIVERSITY OF  KAISERSLAUTERN \hfill  Winterterm 1997/98 \\
   Mathematical Department \hfill November 6, 1997 \\
   Gert-Martin Greuel \hfill Thomas Keilen\\
   \vspace{0.5cm}

   \begin{center}
     \LARGE \bf Introduction to Algebraic Geometry \\
   \end{center}
   \vspace{0.5cm}
   \normalsize
   \parindent0cm
   \textbf{Due date:} Mo, November 24, 1997, 12:00
   \\[0.2cm]

   \setcounter{exercise}{13}
   
   \parindent0cm
   \begin{exercise}
     Let $K$ be an algebraically closed (!) field, and let $V\subseteq\P_K^n$ be 
     a projective algebraic set. $K[V]=K[x_0,\ldots,x_n]/I(V)$ denotes
     the coordinate ring of $V$.\\
     Show that $\dim(K[V])=\mbox{c-}\dim(V)+1$.\\
     \footnotesize Hint, reduce to the case where $V$ is irreducible, and
     connect for ``$\leq$'' the combinatorial dimensions of $V$ and of the affine
     algebraic set $V_i:=\varphi_i(\menge{x_i=1}\cap V)$, where
     $\varphi_i:\menge{x_i=1}\rightarrow \A_K^n$ is the canonical
     homeomorphism. Show moreover that $K[V]_{x_i}\cong
     (K[V_i])[x_i,\frac{1}{x_i}]$, and that for an appropiate choice
     of $i$ the dimension of the localization of $K[V]$ at $x_i$ is just $\dim(K[V])$.
   \end{exercise}

   \begin{exercise}[Veronese Embedding]\label{veronese-embedding}
     Let $K$ be any field. Let
     $M=\menge{m_0,\ldots,m_N}$ be the set of monomials of degree $d>0$
     in $K[X_0,\ldots,X_n]$, and suppose that they are ordered such that
     $m_i=x_i^d$ for $i=0,\ldots,n$ and $m_{n+i}=x_0^{d-1}x_i$ for
     $i=1,\ldots,n$. \\
     We define the \textit{Veronese Embedding} of $\P_K^n$ into $\P_K^N$ by
     \[
        \rho_d:\P_K^n\rightarrow \P_K^N:P\mapsto
        (m_0(P):\ldots:m_N(P)).
     \]
     \begin{enumerate}
       \item Define a ringhomomorphism 
         $\abbil{\theta}{K[Y_0,\ldots,Y_N]}{K[X_0,\ldots,X_n]}$ by
         $\theta(f) = f(m_0,\ldots,m_N)$ and show that $\Ker(\theta)$
         is a homogeneous prime ideal.
       \item Show that for any $i=0,\ldots,N$ and
         $m_i=X_0^{\alpha_0}\cdots X_n^{\alpha_n}$ we have:
         \begin{enumerate}
            \item $Y_i^d-\prod_{j=0}^n Y_j^{\alpha_j}\in\Ker(\theta)$,
            \item $Y_0^{d-1}Y_i-Y_0^{\alpha_0}\prod_{j=1}^n
              Y_{n+j}^{\alpha_j}\in\Ker{\theta}$.
         \end{enumerate}
       \item Show that: $\im(\rho_d)=V(\Ker(\theta))$.\\
         {\footnotesize Hint, given $Q=(Q_0:\ldots:Q_N)\in V(\Ker(\theta))$
           use b.(i) to show that w.\ l.\ o.\ g.\ $Q_0=1$ and use b.(ii) to 
           show that $P=(1:Q_{n+1}:\ldots:Q_{2n})$ maps to $Q$.}
       \item Show finally, that $\rho_d:\P_K^n\rightarrow\im(\rho_d)$ is a
         homeomorphism, where the image carries the subspace topology
         of $\P_K^N$.
     \end{enumerate}
   \end{exercise}

   \begin{exercise}[Veronese Surface]
     Let $K$ be an field of characteristic 0, and
     let $V$ be the Veronese embedding of $\P_K^2$ into $\P_K^5$.\\
     Show that any curve in $V$ is a hypersurface section, i.\ e.\ if
     $C\subset V$ is a pure dimensional projective algebraic set of dimension one, then 
     there exists a hypersurface $H$ in $\P_K^5$ such that $C=H\cap
     V$.\\
     \footnotesize Hint, use exercise \ref{veronese-embedding} d. and 
     the fact that for any polynomial $V(f)=V(f^2)$. Note, the
     condition on the characteristic is not necessary.
   \end{exercise}

\end{document} 
