

\magnification=\magstep1
\def \bA {{\bf A}}
\def \bP {{\bf P}}
\def \bZ {{\bf Z}}
\def \bG {{\bf G}}
\def \bQ {{\bf Q}}
\def \bO {{\bf O}}
\def \bF {{\bf F}}
\def \bT {{\bf T}}
\def \bP {{\bf P}}


{Mathematics 535 \hfill Algebraic Geometry \hfill Fall 2016}

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\centerline{ Problem Set 10 (Last revised \number\month/\number\day/\number\year)}
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\item{1.}{(Harris 7.9) Let $\phi:X --\to\bP^n$ be a rational map with graph
$\Gamma_\phi$.  Show that there is a unique
maximal
open subset $U \subset X$ such that projection on the
first factor induces an isomorphism from
$\pi_1^{-1}(U)\subset \Gamma_\phi$ to U.  Show that there is a pair
$(U,f)$
in the equivalence class $\phi$ such that for all other $(V,g)$ in
the class, $V \subset U$.}

\medskip

\item{2.}{(Harris 7.10) Show that two varieties $X,Y$ are birational
  if and only if they have isomorphic function fields if and only if
  they contain nonempty dense open subsets isomorphic to one
  another. }

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\item{3.}{(Harris 7.13) Show that for all m,n the product $\bP^m\times
  \bP^n$ is rational.  Give an explicit birational isomorphism of
  $\bP^m\times \bP^n$ and $\bP^{m+n}$ (see also 7.28).}



\medskip

\item{4.}{(Harris 7.27) Consider the projection of a quadric
  hypersurface in $Q\subset \bP^n $ from a point $p\in Q$ and assume
  that the rank of $Q$ is $n+1$.  Show that this projection is a
  birational isomorphism of $Q$ with $\bP^{n-1}$ and describe this map
  in terms of blowing up and blowing down.}




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\item{5.}{(Harris 11.9 ) Let $\Gamma \subset \bP^n$ be a general
  collection of d points.  It is known that $\Gamma$ is a complete
  intersection if and only if $n=1,d=1,2$ or $d=4,n=2$.  Show this for
  $n=2, d\leq 7$.}



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\item{6.}{(Harris 11.10) {Let $\Gamma \subset \bP^2$ be a general
    collection of d points. Show that $\Gamma$ is a set-theoretic
    complete intersection.}







\bye



