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   \Blatt{6}
   \Abgabe{01/12/2001}


   \Nr\textbf{[Twisted Cubic]}\\[0.1cm]
   Let $K$ be an algebraically closed field.
   In exercise 5 we have shown that $V=\big\{\big(t,t^2,t^3\big)\in
   \A_K^3\;\big|\;t\in K\big\}$ is an affine algebraic variety in
   $\A_K^3$. Find the vanishing ideals of $V$ and of its projective
   closure $\overline{V}\subset\P_K^3$, where we identify $\A_K^3$
   with the chart $U_0=\{x_0=1\}\subset\P_K^3$.

   \Nr
   Let $K$ be an algebraically closed field, and let
   $V=V\big(x_1^2-x_0x_2\big), V'=V(x_1x_2-x_0x_3), W=V\cap V'\subset\P_K^3$.
   \begin{enumerate}
   \item Calculate $W_\infty$ and $I(W_\infty)$.
   \item Show that $W$ is the union of the twisted cubic and a line.
   \end{enumerate}

   \Nr
   Let $S$ be a graded ring and $I, I'\unlhd S$ be homogeneous
   ideals. Show that:
   \begin{enumerate}
   \item $I+I'$, $I\cdot I'$, $I\cap I'$, $I:I'$ and $\sqrt{I}$ are
     homogeneous ideals. 
   \item $I$ is a prime ideal if and only if $f\cdot g\in I$ with
     $f,g\in S$ homogeneous implies $f\in I$ or $g\in I$.
   \end{enumerate}

   \Nr
   Let $K$ be an algebraically closed field, $V\subseteq \P^n_K$ a
   projective variety. Show that
   $\dim(V)=\max\big\{\dim\varphi_i(V\cap
   U_i)\;\big|\;i=0,\ldots,n\big\}$. 

   {\scriptsize Hint: Show that $K[V\cap U_i]\cong
     K[x_0,\ldots,x_n]/\big(I+(x_i-1)\big)$, and use Krull's principle
     ideal theorem to calculate the dimension of the latter, if
     $x_i-1$ is not a unit modulo $I$.}

   
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