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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 17, 1997, 12:00
   \\[0.2cm]
   
   \setcounter{exercise}{10}
   \parindent0cm
   
   \begin{exercise}
     Let $K$ be an algebraically closed (!) field, $V\subseteq \A_K^n$ an
     irreducible algebraic set of dimension $r$ and let $H$ be a
     hypersurface with $V\not\subseteq H$.\\
     Show that the dimension of any irreducible component of $V\cap H$ 
     has dimension $r-1$, in particular $\dim(V\cap H)=r-1$, if $V\cap 
     H\not= \emptyset$.\\
     \small (Hint, use Krull�s principal ideal theorem and the formula
     connecting the dimension of rings and the height of prime ideals.)
   \end{exercise}
   
   \begin{exercise}
     Let $K$ be an algebraically closed (!) field, $I=\langle f_1,\ldots, f_r\rangle \unlhd K[X_1,\ldots,X_n]$
     and $V=V(I)$.\\
     Show that every irreducible component of $V$ has dimension at 
     least $n-r$, in particular $\dim(V)\geq n-r$, if $V\not=\emptyset$.
   \end{exercise}

   \begin{exercise}
     Let $K$ be algebraically closed and let $I\unlhd
     K[X_0,\ldots,X_n]=K[\underline{X}]$ be a homogeneous ideal, $V(I)\subset\P^n_K$.\\
     Then the following conditions are equivalent:
     \begin{enumerate}
        \item $V(I)=\emptyset$,
        \item $\sqrt{I}=K[\underline{X}]$ or
          $\sqrt{I}=K[\underline{X}]_+:= \bigoplus_{d>0}
          K[\underline{X}]_d$,
        \item $K[\underline{X}]_d\subset I$ for some $d>0$.
     \end{enumerate}
   \end{exercise}

\end{document} 
