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\begin{document} 
 
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   \large UNIVERSITY OF  KAISERSLAUTERN \hfill  Winterterm 1997/98 \\
   Mathematical Department \hfill October 25, 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 3, 1997, 12:00
   \\[0.2cm]
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   \begin{exercise}
     Visualise the affine algebraic sets corresponding to the following
     ideals.
     \begin{enumerate}
        \item $(Y^2-X^2-X^3)\lhd \R[X,Y]$
        \item $(Y^2-ZX^2)\lhd \R[X,Y,Z]$ 
        \item $(XZ)\lhd\R[X,Y,Z]$
        \item $(XZ,XY,YZ)\lhd \R[X,Y,Z]$
     \end{enumerate}
   \end{exercise}

   \begin{exercise}
     Find ideals describing the following affine algebraic sets. \\
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   \end{exercise}

   \begin{exercise}
     Given an infinite field $K$. Let $I=(X^2-Y), J=(XY-1)\lhd K[X,Y]$, and let $V=V(I)$ and $V'=V(J)$.\\
     Show that $I=I(V)$ and $J=I(V')$, and that moreover $K[V]\cong
     K[T]$, while $K[V']\cong K[T,1/T]$.
   \end{exercise}

   \begin{exercise}
     Let $X$ be a topological space, $Y\subset X$ be
     irreducible and $\emptyset\not=U\subset Y$ be open.
     \begin{enumerate}
        \item Show that $U$ is irreducible and dense in $Y$ (where
          dense in $Y$ means $\overline{U}=Y$). 
        \item Show that $\overline{Y}$ is irreducible.
     \end{enumerate}
   \end{exercise}

   \begin{exercise}
     Viewing $\A_\R^1$ as a subset of $\A_\C^1$, show that $\A_\R^1$
     is dense in $\A_\C^1$,
     i.\ e.\ $\R$ is dense in $\C$ in the Zariski-Topology.
   \end{exercise}


\end{document} 
