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{Mathematics 535 \hfill Algebraic Geometry \hfill Fall 2002}

\bigskip


\centerline{ Problem Set 3 (Last revised \number\month/\number\day/\number\year)}
\bigskip

\item{3.2}{Let $\Psi,\Lambda \subset \bP^n$ be complementary linear subspaces
and let $X \subset \Psi, Y\subset \Lambda$ be subvarieties. Show that the union of
all lines joining points of X to points of Y is a variety.}

\itemitem{Proof:}{(J. Tedor)}

\medskip

\item{3.7}{Show that if $\ell$ is any line in $\bP^n$ not meeting a subvariety X 
then there exist homogeneous polynomials $F, G \in I(X)$ with no common zeros
on $\ell$.}
\itemitem{Proof:}{(T. Pham)}



\medskip

\item{3.8}{Find the equations of the projection of the twisted cubic curve
from the point $[1,0,0,1] $ and from $[0,1,0,0]$. Show that any
projection of the twisted cubic from a point is projectively
equivalent to one of these two.}

\itemitem{Proof:}{(K. Dalili)}



\medskip

\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.}

\itemitem{Proof:}{(B. Kennedy)}


\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.}

\itemitem{Proof:}{(S. Ganguly)}



\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}

\itemitem{Proof:}{(S. Ganguly)}

\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.
\itemitem{Proof:}{(E. Henning)
}


\medskip

\item{4.7}{Show that the universal hyperplane section $\Omega_X$ of X does not
admit a rational section in the following cases
$$\cases {(i)\ X \subset \bP^2 &a smooth plane conic \cr
         (ii)\  X \subset \bP^3 & a twisted cubic \cr  }$$
\itemitem{Proof:}{(J. Tedor)}


\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.}

\itemitem{Proof:}{(T. Pham)}

\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).}

\itemitem{Proof:}{(K. Dalili)}


\medskip


\item{4.12}{(a) For any n, find a rational section of the universal
hyperplane $\Gamma \subset (\bP^N)^*\times \bP^n$. (b) For n odd, find a section of 
$\Gamma$. (c) For n even show that there does not exist
a section of $\Gamma$.}
\itemitem{Proof:}{(B. Kennedy)
}


\medskip


\item{4.13}{Show that the universal plane conic ${\cal X} \subset 
\bP^5\times \bP^2$ does not admit a rational section.}

\itemitem{Proof:}{(S. Ganguly)}



\medskip

\item{4.14}{(a) Let ${\cal X} \subset B \times \bP^3$ be any 
family of twisted cubics. Show that ${\cal X} $ admits a rational 
section. (b) Find a family of rational normal curves of degree 4 which does not have
a rational section.}

\itemitem{Proof:}{(E. Henning)}
\medskip


\item{4.16}{Let $U\subset \bP^n\times \bP^n$ be the complement of the
diagonal.  Let $\Omega=\{(p,q,r): r\in \overline{pq}\}\subset U \times \bP^n$.
Show that $\Omega $ is a subvariety of $U\times \bP^n$. More generally show that
for any k the subset $U \subset (\bP^n)^k$ formed by the points
$(p_1,\ldots,p_k)$ such that $p_1,\ldots,p_k$ are linearly independent is open and
the set $$\Omega=\{(p_1,\ldots,p_n;r): r\in \overline{p_1\ldots p_n}\}\subset U \times \bP^n$$
is a subvariety. Find the closure of U in $(\bP^n)^k\times \bP^n$.}


\itemitem{Proof:}{(J. Burdges)}


\medskip

\item{4.17}{Let $V \subset ((\bP^n)^*)^{n-k}$ be the open set of
(n-k) tuples $(H_1,\ldots,H_{n-k})$ for which $H_i$ are linearly independent hyperplanes in V.
Let $\Xi= \{(H_1,\ldots,H_{n-k};r): r \in H_1\cap\ldots\cap H_{n-k}
\subset V \times \bP^n$.  Show that $\Xi$ is a family of k--planes.
When $n=3, k=1$ is this variety isomorphic to one constructed
in 4.16?}


\itemitem{Proof:}{(J. Burdges)}



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