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

\bigskip


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

\item{2.18}{Let $C_{\alpha,\beta}$ be the curve of problem 1.29 (a
so-called rational quartic curve).  Show that $C_{\alpha,\beta}$ lies
on the Segre surface $S$ which is the zero set of $Z_0Z_3-Z_1Z_2$ in
$\bP^3$.  Show that $S$ is the unique quadric surface containing
$C_{\alpha,\beta}$, and that $C_{\alpha,\beta}$ is the zero set of a
bihomogeneous polynomial of bidegree $(1,3)$ on $S \simeq \bP^1\times
\bP^1$.  Use this to give an alternate solution to problem 1.29.}


\medskip
\item{2.19}{Use 2.18 to show that there is a continuous family of curves 
$C_{\alpha,\beta}$ not projectively equivalent to one another.}

\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 $\Omega$ in
$(\bP^n)^k\times \bP^n$.}


\medskip
\item{5.2}{Show that he following are equivalent for homogeneous
ideals $I, J \subset K[Z_0,\ldots , Z_n]$
$$\cases {(i)& $ I$  and $J$  have the same saturation\cr
        (ii)& $I_m=J_m$ for all $m$ sufficiently large\cr
        (iii)& $I,J$ generate the same ideal in each localization $K[Z_0,\ldots,
Z_n,
Z_i^{-1}]$\cr}$$
}

\medskip

\item{5.13}{Show that for any $d, n\leq (d+1)(d+2)/2$ a general set of
n points in $\bP^2$ imposes independent conditions on curves of degree
d. This means that the space of polynomials vanishing at the points
has codimension n in the space of all homogeneous polynomials of
degree d.  In case $n \leq 2d+1$ what open set in $(\bP^2)^n$ is
implicitly referred to here.}

\medskip
\item{5.14}{Let $C \subset \bP^n$ be a curve of degree d, that is the
zero set of a homogeneous polynomial $F(Z_0,Z_1,Z_2)$ of degree d,
which has no repeated factors in its unique factorization into
irreducibles.  Show that the general line in $\bP^2$ will intersect C
in d points.}


\medskip
\item{6.2}{Suppose that the characteristic of K is not 2.
Show that a vector $\omega \in \bigwedge^2 V$ is decomposable if
and only if $\omega \wedge \omega=0$ and hence the Grassmannian G(2,V) of
2 planes in V  is cut out by quadrics in $\bP(\bigwedge^2V)$.}






\bye
