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   \Blatt{11}
   \Abgabe{28/01/2002}

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

   \Nr
   Let $X$ and $Y$ be quasi-projective varieties. Show that the
   projections 
   $\pi_X:X\times Y\rightarrow X$ and $\pi_Y:X\times Y\rightarrow Y$
   are open, i.~e.~the image of an open set is open.

   {\scriptsize Hint: Note that any open subset of $X\times Y$ is covered by basic open
     sets of the form $D(f)=(X\times Y)\setminus V(f)$, with $f\in
     K[\underline{x},\underline{y}]$ bihomogeneous, and show that the
     image of $D(f)$  is again the union of basic open subsets.}

   \Nr
   Let $X$ and $Y$ be irreducible quasi-projective varieties. Show
   that $X\times Y$ is irreducible.

   {\scriptsize Hint, consider first the case $X=\{x\}$ applying the
     Segre embedding, and apply Exercise 37 to the general case. For
     this you should express the irreducibility with open subsets
     rather than with closed ones.}

   \Nr
   Let $X$, $Y$ and $S$ be quasi-projective varieties and let
   $f:X\rightarrow S$ and $g:Y\rightarrow S$ be morphisms. Show that the
   fibre product $X\times_S Y$ exists.

   {\scriptsize Hint, do the proof in several steps. Consider first
     the case that $X, Y$ and $S$ are affine. Reduce the case that $S$
     is affine, but $X$ and $Y$ are arbitrary to this case. Let then
     $S$ be arbitrary, but $X$ affine, and consider finally the general
     case. It might be worth taking Exercises 26 and 31a into account.}

   
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