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{Mathematics 535 \hfill Algebraic Geometry \hfill Fall 2008}

\bigskip


\centerline{ Problem Set 7A (Last revised \number\month/\number\day/\number\year)}
\bigskip


\item{13.6}{Find the Hilbert function of the Segre variety
$\Sigma_{n,m}=
\sigma(\bP^n\times \bP^m) \subset \bP^{nm+n+m}$ and verify that the
dimension
is $n+m$.}

\medskip

\item{13.8}{Determine the arithmetic genus of (i) a pair of skew lines
in $\bP^3 $ (i) a pair of incident lines in either $\bP^2$ or $\bP^3$
(iii) three concurrent but not coplanar lines in $\bP^3$ and (iv)
three concurrent coplanar lines in either $\bP^2$ or $\bP^3$.}

\medskip

\item{13.9}{Consider a plane curve $X \subset \bP^2$ of degree d and
its image $Y=\nu_2(X) \subset \bP^5$ under the quadratic Veronese map.
Compare the Hilbert polynomials of the two and observe in particular
that the arithmetic genus is the same.}

\medskip


\item{13.17}{Prove a weak form of the Bezout theorem in $\bP^2$: if
$F,G$ are polynomials of degree $d,e$ on $\bP^2$ without common
factors such that $F,G$ generate the ideal of their intersection, then
the intersection consists of $d\cdot e$ points.  Similarly show that
if $\Gamma \subset \bP^3$ is a complete intersection of surfaces of
degrees $d,e,f$ then $\Gamma$ consists of $d\cdot e \cdot f$ points.}


\medskip

\item{13.18}{Find the Hilbert polynomial of a complete intersection in
$\bP^3$ of surfaces of degrees $d,e$.  What is the arithmetic genus of
this complete intersection?}


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