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   \Blatt{7}
   \Abgabe{18/12/2001}

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

   \Nr\textbf{[Veronese Embedding]}\\[0.1cm]
   Let
   $M=\{m_0,\ldots,m_N\}$, $N=\binomial{d+n}{n}-1$ 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
   \[
   \rho_d:\P_K^n\rightarrow \P_K^N:p\mapsto
   \big(m_0(p):\ldots:m_N(p)\big).
   \]
   \begin{enumerate}
   \item Define a ringhomomorphism 
     $\theta:K[y_0,\ldots,y_N]\rightarrow K[x_0,\ldots,x_n]$ 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: $\im(\rho_d)=V\big(\Ker(\theta)\big)$, i.~e.~$\im(\rho_d)$
     is a closed subvariety of $\P_K^N$.

     {\footnotesize Hint, given $q=(q_0:\ldots:q_N)\in V\big(\Ker(\theta)\big)$
       use b.(a) to show that w.\ l.\ o.\ g.\ $q_0=1$ and use b.(b) 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 an
     isomorphism. 

     {\footnotesize Hint, use Exercise 26 and cover $V=\im(\rho_d)$
       and $W=\P_K^n$ by $n+1$ suitable open subvarieties $V_i$
       respectively $W_i$, where the $W_i=\{x_i=1\}$ are actually affine, and
       then consider w.~l.~o.~g.~only the case $i=0$.} 
   \end{enumerate}
   
   \Nr
   Let $V$ and $W$ be quasi-projective varieties over $K$ and
   $\varphi:V\rightarrow W$ be any map. Suppose that $\{V_i\;|\;i\in I\}$ is 
   an open covering of $V$ and $\{W_i\;|\;i\in I\}$ is
   an open covering of $W$ such that $\varphi(V_i) \subseteq W_i$
   for all $i\in I$.

   Then the following are equivalent:
   \begin{enumerate}
   \item $\varphi$ is a morphism.
   \item $\varphi_{|}:V_i\rightarrow W_i$ is a morphism for all
     $i\in I$.
   \end{enumerate}
   If moreover $W_i\subseteq \A_K^{n_i}$ is quasiaffine, these are equivalent
   to: 
   \begin{enumerate}\setcounter{enumi}{2}
   \item $\varphi^*\ko(W_i)\subseteq\ko(V_i)$ for all $i\in I$,
     where $\varphi^*(h)=h\circ \varphi$ for $h\in\ko(W_i)$.
   \item For all $i\in I$ we have
     $\varphi^*(x_{i,j})$ is regular for all $j=1,\ldots,n_i$
     (where $x_{i,j}$ is the j-th coordinate function of $\A^{n_i}_K$).
   \end{enumerate}
   {\footnotesize Hint, prove first the equivalence of (a) and (b),
     then you may for the remaining part assume that $V=V_i$ and
     $W=W_i$. (d) $\Rightarrow$ (b) is hard!}
   

   
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