

\magnification=\magstep1
\def \bA {{\bf A}}
\def \bP {{\bf P}}
\def \bZ {{\bf Z}}
\def \bG {{\bf G}}
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\def \bF {{\bf F}}
\def \bT {{\bf T}}
\def \bP {{\bf P}}


{Mathematics 535 \hfill Algebraic Geometry \hfill Fall 2016}

\bigskip


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


Recall that a morphism $\phi:X \rightarrow Y$ for projective
varieties $X,Y$ is a continuous function such that for each open set
$U$ of $Y$ and regular function $f$ on $U$ the composition $f\circ
\phi$ is regular on $\phi^{-1}(U)$.
\medskip
Let $K$ be an algebraically closed field.
\bigskip

\item{1.}{Verify that if $X\subset \bP^m$ is a projective variety
and $F_i(x_0,\ldots,x_m), i=0,\ldots, n$ are homogeneous polynomials
of degree $d$ with no common zero on $X$ then the function \break
$\phi([x_0,\ldots,x_m])
=[F_0(x_0,\ldots,x_m),\ldots,F_n(x_0,\ldots,x_m)]$ is a well defined
function from $X$ to $\bP^n$.  By considering the restriction of
$\phi$ to the open subsets of the affine variety $X \cap
\{[x_0,\ldots, x_m] | x_j \not = 0\}$ where $F_i(x_0,\ldots,n_m) \not
= 0$ show that $\phi$ is a morphism from $X$ to $\bP^n$ (see the
middle paragraph on page 21 of Harris).}
\medskip

\item{2.}{The goal of this problem is to
determine the set of morphisms from $X=\bP^1$ to $Y=\bP^1$.  Take
homogeneous coordinates $[x_0,x_1]$ on $X$ and $[s,t]$ on $Y$.}

\itemitem{a)}{Suppose that r(z) is a rational function of the variable
z, that is a member of the fraction field $K(z)$ of the polynomial
ring
$K[z]$ ($K(z)$ is the localization of $K[z]$ with respect to the
multiplicative set of nonzero polynomials).  If $r(z)=f(z)/g(z)$ is an
expression for the rational function where $f(z),g(z)$ have no common
root, define the degree d of $r(z)$ to be the maximum of degrees of
$f(z),g(z)$.  Show that the rational functions 
$F=x_1^df(x_0/x_1),
G=x_1^dg(x_0/x_1)$
are homogeneous polynomials in $x_0,x_1$ of degree
d,
which factor as the product of d linear polynomials, and have no
common
zeros on $P^1$.  Let $\phi_r([x_0,x_1])=[F(x_0,x_1),
G(x_0,x_1)]$ be the morphism from $\bP^1$ to $\bP^1$ defined in
problem 1.


\itemitem{b)}{Show that the construction of (a) gives a one to one map
of the monoid of rational functions under composition to the monoid of
morphisms from $\bP^1$ to $\bP^1$.}

\itemitem{c)}{Show that every morphism $\phi$ from $\bP^1$ to itself
  arises
from the construction in (a).  Hint: on some nonempty open set in $X$
show that $\phi([x_0,x_1])$ is a rational function $r$ of $x_1/x_0$
and verify that $\phi=\phi_r$.

\itemitem{d)}{Use the preceeding to show that the group of
automorphisms $Aut(\bP^1)$ is isomorphic to the group of degree 1
rational functions under composition and that this is isomorphic to
$PGL(2,K)$.}

\medskip
\item{3.}{(Harris 2.8) Show that the image $Y=\nu_d(X)\subset \bP^N$ of a
  projective variety
$X\subset \bP^n$ is isomorphic to X via the Veronese map.}





\medskip

\item{4.}{(Harris 2.9) Use 2.8 to show that any projective variety is isomorphic
  to the
intersection of a Veronese variety and a linear subspace, and hence
is isomorphic to an intersection of quadrics.}


\medskip

\item{5.}{(Harris 2.10) Let  $Y=\nu_d(X)\subset \bP^N$ be the Veronese image
of a projective variety $X\subset \bP^n$. What is the relation of the
homogeneous
coordinate rings? Show that the homogeneous coordinate ring is not an
isomorphism
invariant of a projective variety, and construct varieties in a
projective
space that are isomorphic but not projectively equivalent.}

\medskip

\item{6.}{(Harris 2.13) Show that the twisted cubic in $\bP^3$ is the intersection
  of the
Segre threefold
with a three plane in $\bP^5$.}

\medskip

\item{7.}{(Harris 2.15) Show that the image of the diagonal $\Delta \subset
  \bP^n\times \bP^
n$
under the Segre map is the Veronese variety $\nu_2(\bP^n)$ lying in a
subspace
of $\bP^{n^2+2n}$.  Conclude that in general the diagonal
$\Delta_X\subset X \times X$ is a subvariety of the product variety,
and
similarly for the diagonal in the n--fold product $X^n$.}

\medskip

\item{8.}{(Harris 2.24) Let $X\subset \bP^n$ be a projective variety and let
  $\phi: X \to \b
P^m$ be any regular map. Show that the graph $\Gamma_\phi \subset
X\times \bP^n
\subset \bP^n\times \bP^m$
is a subvariety.}


\bye


