%
%



\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 4B (Last revised \number\month/\number\day/\number\year)}
\bigskip

\item{3.9}{Show that the curves of exercise 1.29 may be realized as
projections of a rational normal curve in $\bP^4$ from points
$p_{\alpha,\beta}\in
\bP^4$ and
use these to show that the projections $\pi_p(X),\pi_{p'}(X)$ of a
variety X from different points need not be projectively equivalent.}

\medskip


\item{4.2}{Let $X\subset \bP^n$ be a projective variety and let
$\{V_b\}$ be a  family of projective varieties in $\bP^n$ with
base B. Show that the set
$$\{b \in B: X\cap V_b \not = \emptyset\}$$
is closed in B. In general, if $\{W_b\}$ is another family of
projective
varieties in $\bP^n$ with base B show that the set $\{b:V_b\cap W_b
\not = \emptyset\}$
is a closed subvariety of B.}

\medskip

\item{4.3}{With the notation of 4.2, let $\{W_c\}$ be another family
of
projective varieties in $\bP^n$ with base C. Show that the set
$$\{(b,c):V_b\cap W_c \not = \emptyset\}$$ is a closed subvariety of
$B \times C$, and show that this implies the statements in exercise
4.2}

\medskip



\item{4.4}{ Show that for each subvariety $X \subset \bP^n$ and any
family
$\{V_b\}$ of subvarieties in $\bP^n$ the subset $\{b \in B: X\subset
V_b\}$
is closed in B.

\medskip

\item{4.10}{Consider $\bP^N$ as the parameter space of hypersurfaces
of degree d in $\bP^n$, where $N={n+d \choose d}-1$. Show that the
subset $\Sigma \subset \bP^N$ consisting of nonprime polynomials
$F(X_0,\ldots,X_N)$ is a projective subvariety of $\bP^N$. In case
$d=2,n=2$ identify this subvariety.  We will show later that in
general it is the collection of hypersurfaces of degree d containing a
hypersurface of strictly smaller degree.}

\medskip

\item{4.11}{Let $X \subset \bP^n$ be any hypersurface of degree d
given by a homogeneous polynomial $F(Z_0,Z_1,\ldots,Z_n)$. Show that
the subset of hyperplanes $H \subset \bP^n$ such that the restriction
of F to H factors is a subvariety of $\bP^n$ (hence the set of
hyperplanes H such that $H \cap X$ contains a hypersurface of degree
$< d$ form a subvariety).}









\bye
