

%
%



\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 6B (Last revised \number\month/\number\day/\number\year)}
\bigskip




\item{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 $n=2, d\leq 7$.}


\medskip

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


\medskip

\item{11.25}{Show that if $X\subset \bP^n$ is an irreducible curve
then the chordal variety $S(X)$ is three-dimensional unless $X$ is
contained in a plane.}

\medskip

\item{11.29}{Let $M$ be the projective space of $m\times n$ matrices,
and let $M_k\subset M$ be the subvariety of matrices of rank at most
$k$. Take $k$ so that $2k < \min(m,n)$. Show that the chordal variety
$S(M_k)$ is equal to the subvariety $M_{2k} \subset M$ of matrices of
rank at most $2k$.  Indeed.  show that for a map $A:K^m \to K^n$ of
rank $2k$ and any pair of complementary $k$-dimensiaaonal subspaces
$\Lambda,\Xi\subset {\rm \ image}M(A)$ the composition of A with
projections of $\Lambda$ and $\Xi$ give an expression of A as a sum of
two matrices of rank k. Deduce that in the case $\bP^n=M, X=M_k$ the
fiber of the incidence correspondence $\Sigma$ in 11.22 over a general
point $S(X)$ has dimension $2k^2$ and hence that $\dim(S(M_k)) \leq 2
\dim(M_k)+1-2k^2$.  }

\medskip



\bye


























