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   \large UNIVERSITY OF  KAISERSLAUTERN \hfill  Winterterm 1997/98 \\
   Mathematical Department \hfill 5th 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:} 19th January, 1998 
   \\[0.2cm]
   
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   \setcounter{exercise}{33}

   \begin{exercise}
     Let $K$ be algebraically closed, $X$ and $Y$ be varieties over
     $K$, $\pi_X:X\times Y\rightarrow X$ the projection.\\
     Is $\pi_X$ open? Is it closed? Prove or find counterexamples.
   \end{exercise}

   \begin{exercise}\label{tensor}
     Let $R$ be a ring, $A$ and $B$ be $R$-algebras and $A$ be free
     as an $R$-module with free generators $\menge{a_i|i\in
       I}$, i.\ e.\ $A=\bigoplus_{i\in I}R\cdot a_i$ and for any 
     $R$-module $M$ with $m_i\in M$ ($i\in I$) there is exactly one
     $R$-linear map $\alpha:A\rightarrow M$ with
     $\alpha(a_i)=m_i$. \\
     Show that $A\otimes_R B$ is free as a $B$-module with free
     generators $\menge{a_i\otimes 1|i\in I}$.
   \end{exercise}

   \begin{exercise}
     Let $K$ be algebraically closed and $X$ and $Y$ be varieties over 
     $K$. Show that 
     \begin{displaymath}
       \begin{array}{ccc}
         \ko(X)\otimes_K\ko(Y) & \cong & \ko(X\times Y)\\
         g\otimes h & \mapsto & g\cdot h.
       \end{array}
     \end{displaymath}
     {\small (Hint, for surjectivity proceed in three steps. Step one,
       both $X$ and $Y$ are affine, then this has been shown in the
       lecture. Step two, $X$ is affine and $Y=Y_1\cup\ldots\cup Y_r$
       with $Y_i$ affine. Then use exercise \ref{tensor} and step 1 to show that
       given any $f\in\ko(X\times Y)$, $f_|\in\ko(X\times Y_i)$  has 
       in some sense a unique
       decomposition of the form $\sum_{j\in J} g_j\otimes h_{ij}$
       with $g_j\in \ko(X)$ and $h_{ij}\in\ko(Y_i)$, where the $h_{ij}$ 
       glue to global regular functions $h_j\in\ko(Y)$. Step three,
       reduce the general case in the same manner to Step 2.)}
   \end{exercise}

   \begin{exercise}[Quadratic Cremona Transformation]
     Define the quadratic Cremona transformation
     $\varphi:\P_K^2\dashrightarrow\P_K^2:(x_0:x_1:x_2)\mapsto
     (x_1x_2:x_0x_2:x_0x_1)$.\\
     Find open and dense subsets $U$ and $V$ in $\P_K^2$ such that
     $\varphi_|:U\rightarrow V$ is an isomorphism, i.\ e.\ show that
     $\varphi$ is birational. What is the definition set of $\varphi$?
   \end{exercise}

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