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

\bigskip


\centerline{ Problem Set 5B (Last revised \number\month/\number\day/\number\year)}
\bigskip




\medskip
\item{6.4}{For any point $p \in \bP^3$ and any plane
$H \subset \bP^3$ containing p let $\Sigma_{p,H} \subset \bG(1,3)$ be
the set of lines in $\bP^3$ passing through p and lying in H.  Show
that
under the Plucker imbedding $\bG(1,3) \rightarrow \bP^5$ the
subvariety $\Sigma_{p,H}$ is
mapped to a line, and conversely every line in $\bP^5$ lying on
$\bG(1,3)$ is
of the form $\Sigma_{p,H}$ for some p and H.}

\medskip

\item{6.5}{For a point $p \in \bP^3$ let $\Sigma_p \subset \bG(1,3)$
be the set of lines in $\bP^3$ passing through p.  For any plane
$H \subset \bP^3$ let $\sigma_H \subset \bP^3$ be the set of lines
in $\bP^3$ contained in H.  Show that the Plucker embedding carries
both $\Sigma_p, \Sigma_H$ into 2--planes in $\bP^5$.  Show that
conversely any 2--plane $\Lambda\simeq \bP^2 \subset \bG(1,3)
 \subset \bP^5$ either equals $\Sigma_p$ for some p or $\Sigma_H$
for some H. }
\medskip

\item{6.6}{Let $\ell_1,\ell_2 \subset \bP^3$ be skew lines (that is
nonintersecting lines).  Show that the set $Q\subset \bG(1,3)$ of
lines in $\bP^3$ meeting both is the intersection of $\bG(1,3)$ with a
three--plane $\bP^3\subset
\bP^5$ and hence is a quadric surface. Deduce that $Q$ is isomorphic
to $\bP^1\times\bP^1$.
Do the same problem for lines $\ell_1,\ell_2 \subset \bP^3$ which
meet.}



\medskip

\item{6.8}{Let $\Sigma_{1,k} \simeq \bP^1\times \bP^k \subset
\bP^{2k+1}$
be the Segre variety, and let $\Lambda_p$ be the fiber over $p \in
\bP^1$.
Show that $\Lambda_p$ is a k-plane in $\bP^{2k+1}$ and that the
assignment $p \mapsto \Lambda_p$ gives a regular map of $\bP^1$ to the
Grassmannian $\bG(k,2k+1)$ with image a rational normal curve lying in
a
(k+1)-plane in $\bP(\bigwedge^{k+1}K^{2k+2})$.}


\medskip




\item{6.11}{Let $\Lambda \subset \bP^n$ be a k-plane and let
$I(\Lambda)_d$ be the graded piece of degree d in the homogeneous
ideal of all homogeneous polynomials vanishing on $\Lambda$. Show that
the codimension of $I(\Lambda)_d$ in the vector space $S_d$ of all
homogeneous
polynomials of degree d is ${k+d \choose d}$.  Show that the map 
$$\nu_d:G(K,n) \to G({k+d\choose d},{n+d\choose d})$$ obtained by
associating to a k-plane the subspace of linear functionals
on $S_d$ which vanish on $I(\Lambda)_d$ is regular.}

\medskip

\item{6.20}{Let $Q\subset \bP^3$ be the zero set of $Z_0Z_3-Z_1Z_2$.
Show that the Fano variety $F_1(Q)$ of all lines contained in Q is a union
of two conic curves. Compare this with the parametric description
of the Fano variety given by using the Segre map.}



\bye