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   \large UNIVERSITY OF  KAISERSLAUTERN \hfill  Winterterm 1997/98 \\
   Mathematical Department \hfill 3rd January, 1998\\
   Gert-Martin Greuel \hfill Thomas Keilen\\
   \vspace{0.5cm}

   \begin{center}
     \LARGE \bf Introduction to Algebraic Geometry \\
   \end{center}
   \vspace{0.5cm}
   \normalsize
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   \textbf{Due date:} 12th January, 1998 
   \\[0.2cm]
   
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   \setcounter{exercise}{29}

   \begin{exercise}
     Let $\ko(U)$ be a $K$-Algebra of $K$-valued functions on a
     topological space $U$, containing the constant functions. Show that
     the following are equivalent:
     \begin{enumerate}
        \item $D(f)=\menge{P\in U|f(P)\not=0}$ is open in $U$ for all
          $f\in\ko(U)$,
        \item $f$ is continous for every $f\in\ko(U)$, where
          $K\hat{=}\A_K^1$ carries the Zariski topology. 
     \end{enumerate}
   \end{exercise}

   \begin{exercise}
     Let $X$ be a quasi projective variety in the sense of chapter
     I.4, i.\ e.\ $X$ is isomorphic to an open subset of a projective
     algebraic set. \\
     Show that $X$ is a variety in the sense of chapter I.5.\\
     {\small (Hint, use exercise 28 and lemma I.5.6.)}
   \end{exercise}

   \begin{exercise}[Universal Property of Products]

     Let $K$ be an algebraically closed field, and $X$ and $Y$ be
     spaces with functions over $K$. $\pi_X:X\times Y\rightarrow X$ denote the
     projection to $X$ and $\pi_Y:X\times Y\rightarrow Y$ the
     projection to $Y$.\\
     Show that $\pi_X$ and $\pi_Y$ are morphisms, and that for any
     space with functions $Z$ the map 
     \[
        \begin{array}{ccc}
           \Mor(Z,X\times Y)&\rightarrow
           &\Mor(Z,X)\times\Mor(Z,Y)\\
           \varphi & \mapsto &
           (\pi_X\circ\varphi,\pi_Y\circ\varphi) 
        \end{array}
     \]
     is a bijection. \\
     {\small (Remark, the last statement is the universal property of
       the product of spaces with functions and characterizes the product up to
       isomorphy. )}

   \end{exercise}

   \begin{exercise}
     Let $K$ be an algebraically closed  field. Show that the
     following statements are equivalent:
     \begin{enumerate}
        \item $V\subseteq \P^r_K\times\P^s_K$ is closed,
        \item there exist bihomogeneous polynomials
          $f_1,\ldots,f_k\in\K[X_0,\ldots,X_r,Y_0,\ldots,Y_s]$
          such that $V=V(f_1,\ldots,f_r)$, where a
          polynomial $f$ is said to be bihomogeneous, if it is
          homogeneous as a polynomial in $\underline{X}$ with
          coefficients in $K[\underline{Y}]$ and also as a polynomial
          in $\underline{Y}$ with coefficients in $K[\underline{X}]$.
     \end{enumerate}
     {\small (Hint, for a. to b. use the Segre embedding, and for
       b. to a. cover $\P^r_K\times\P^s_K$ by affine varieties.)}
   \end{exercise}

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