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   \Blatt{10}
   \Abgabe{21/01/2002}

   Throughout this example sheet $K$ will be an algebraically closed field.

   \Nr
   We 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 $\varphi$?

   \Nr
   Show that $\P_K^1\times\P_K^1$ is birationally equivalent to
   $\P_K^2$, i.~e.~there is a birational map
   $\pi:\P_K^1\times\P_K^1\dashrightarrow \P_K^2$.

   {\scriptsize Hint, via the Segre embedding $\P_K^1\times\P_K^1$ is
     just the quadric from Exercise 30. Consider the projection
     $\pi_q:\P_K^3\dashrightarrow \P_K^2$ from the point $q=(1:0:0:0)$.}
   
   \Nr
   Let $\alpha,\beta\in K$. 
   \begin{enumerate}
   \item Show that the rational map
     \begin{displaymath}
       \nu_{\alpha,\beta}:\P_K^1\dashrightarrow\P_K^3: 
       (x_0:x_1)\mapsto \big(x_0^3\cdot(x_0-\beta x_1):x_0^2x_1\cdot(x_0-\beta x_1):
       x_0x_1^2\cdot(\alpha x_0-x_1):x_1^3\cdot(\alpha x_0-x_1)\big)
     \end{displaymath}
     defines actually a morphism.
   \item Show that $\im(\nu_{\alpha,\beta})$ is a closed subvariety of
     $\Sigma_{1,1}$.
   \item Find a bihomogeneous polynomial $g\in K[x_0,x_1,y_0,y_1]$
     of bidegree $(1,3)$ respectively $(1,2)$ such that via the Segre embedding
     $\sigma:\P_K^1\times\P_K^1\rightarrow \Sigma_{1,1}$ the closed
     subvariety $V(g)$ is maped onto $\im(\nu_{\alpha,\beta})$.
   \end{enumerate}
   {\scriptsize Hint, to find the polynomial in c.~it may be worth
     considering $\im(\nu_{\alpha,\beta})$ in local charts. You may,
     without any proof, assume that, if $g$ is an irreducible
     polynomial, then $V(g)$ is irreducible and has dimension one -
     instead you could also use Singular to convince yourself that
     $V(g)$ is irreducible of dimension one.}
   
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