

\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 2016}

\bigskip


\centerline{ Problem Set 11 (Last revised \number\month/\number\day/\number\year)}


\item{1.}{(Harris 11.11) Find a homogeneous quadratic polynomial $Q(Z_0,\ldots,Z_3)$
and a homogeneous cubic polynomial $P(Z_0,\ldots,Z_3)$ whose common zero locus
is a rational normal curve $C \subset \bP^3$.}

\medskip

\item{2.}{(Harris 11.15) Let $X, Y$ be irreducible projective
  varieties, $f:X \to Z$ and $g:Y\to Z$ surjective maps. Show that the
  dimension of the fiber product satisfies
$${\dim}(X\times_Z Y) \geq \dim(X)+\dim(Y)-\dim(Z).$$ Find an example
  where strict inequality holds. Show by example that some irreducible
  components of $X\times_Z Y$ may have strictly smaller dimension.}



\medskip

\item{3.}{(Harris 11.20) Let $X\subset \bP^n$ be an irreducible
  non-degenerate $k$-dimensional variety.  For $l<n-k$ find the
  dimension of the closure of the locus of $l$-planes meeting $X$ in
  at least two points. Show by example that no analogous formula
  exists if we replace two by three, even if we require $l \geq 2$. }


\medskip

\item{4.}{(Harris 11.21) Let $X\subset \bP^n$ be an irreducible
  non-degenerate $k$-dimensional variety. Show that the plane spanned
  by $n-k+1$ general points of $X$ meets $X$ in only finitely many
  points and use this to compute the dimension of the closure of the
  locus of $l$-planes spanned by their intersections with $X$ when $l
  \leq n-k$.  }


\medskip





\item{5.}{(Harris 11.25) Show that if $X\subset \bP^n$ is an irreducible curve
  then the chordal(or secant) variety $S(X)$ is three-dimensional
  unless $X$ is contained in a plane.}





\medskip



\item{6.}{(Harris 11.31) Let $X\subset \bP^n$ be an irreducible nondegenerate
  variety of dimension k. use 11.21 to deduce that for any $l \leq
  n-k$ the general fiber of the secant plane map
$$s_l:X^{l+1} \to \bG(l,n)$$
is finite.}


\medskip


\item{7.}{(Harris 11.39) Show that if $m > n$ there does not exist a
  nonconstant regular map $\phi:\bP^m\to \bP^n$ by using 11.38.Hint:
  First try the case n=1 and look at dimensions of inverse images of a
  hyperplane and a point not on it.  }

\medskip

\item{8.}{(Harris 11.41) Find the dimension of the flag variety
  $\bF(a_1,\ldots,a_n)$ in general.}


\bye













































\item{1.}{(Harris 7.9) Let $\phi:X --\to\bP^n$ be a rational map with graph
$\Gamma_\phi$.  Show that there is a unique
maximal
open subset $U \subset X$ such that projection on the
first factor induces an isomorphism from
$\pi_1^{-1}(U)\subset \Gamma_\phi$ to U.  Show that there is a pair
$(U,f)$
in the equivalence class $\phi$ such that for all other $(V,g)$ in
the class, $V \subset U$.}

\medskip

\item{2.}{(Harris 7.10) Show that two varieties $X,Y$ are birational
  if and only if they have isomorphic function fields if and only if
  they contain nonempty dense open subsets isomorphic to one
  another. }

\medskip

\item{3.}{(Harris 7.13) Show that for all m,n the product $\bP^m\times
  \bP^n$ is rational.  Give an explicit birational isomorphism of
  $\bP^m\times \bP^n$ and $\bP^{m+n}$ (see also 7.28).}



\medskip

\item{4.}{(Harris 7.27) Consider the projection of a quadric
  hypersurface in $Q\subset \bP^n $ from a point $p\in Q$ and assume
  that the rank of $Q$ is $n+1$.  Show that this projection is a
  birational isomorphism of $Q$ with $\bP^{n-1}$ and describe this map
  in terms of blowing up and blowing down.}




\medskip

\item{5.}{(Harris 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{6.}{(Harris 11.10) {Let $\Gamma \subset \bP^2$ be a general
    collection of d points. Show that $\Gamma$ is a set-theoretic
    complete intersection.}







\bye



