

%
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\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 2008}

\bigskip


\centerline{ Problem Set 6A (Last revised \number\month/\number\day/\number\year)}
\bigskip

\item{7.28}{Generalize 7.11 by showing that $\bP^m\times \bP^n$ is birational
with $\bP^{m+n}$ via the map
$$\phi([Z_0,\ldots,Z_n],[W_0,\ldots,W_n])=[Z_0W_0,Z_1W_0,\ldots,Z_mW_0,Z_0W_1,\ldots,
Z_0W_n].$$ What is the graph of this map? Describe the map in terms of
blowing up and blowing down.}

\medskip

\item{11.11}{Find a homogeneous quadratic polynomial $Q(Z_0,\ldots,Z_3)$
and a homogeneous cubic polynomial $P(Z_0,\ldots,Z_3)$ whose common zero locus
is a rational normal curve $C \subset \bP^3$.}

\medskip 

\item{11.39}{Show that if $m > n$ there does not exist a nonconstant
regular map $\phi:\bP^m\to \bP^n$ by using 11.38.}



\medskip

\item{11.44}{Let $X \subset \bP^n$ be an irreducible $k$-dimensional 
variety. Find the dimension
of the universal $(n-l)$-plane section $\Omega^{(l)}(X)$ 
and show that it is irreducible.}

\medskip

\item{12.17}{Observe that a curve of type $(1,d-1)$ on a smooth quadric
$Q \subset \bP^3$ is a rational curve of degree $d$, that is the image
of $\bP^1 $ under a map to $\bP^3$ given by a quadruple of polynomials
of degree $d$ on $\bP^1$, or equaivalently a projection of the
rational normal curve in $\bP^d$. Use a count of parameters to show
that for $d \geq 5$ the general rational curve of degree $d$ is not a
curve of type $(1,d-1)$ on a quadric.}





\bye


























