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

\bigskip


\centerline{ Problem Set 3A (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
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 $F=x_0^df(x_1/x_0),
G=x_0^dg(x_1/x_0)$ 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{1.27}{Show that the images of the maps $\mu,\nu :\bP^1 \to
\bP^2$
given by $\mu[x_0,x_1]=[x_0^3,x_0x_1^2,x_1^3]$ and
$\nu[x_0,x_1]=[x_0^3,x_0x_1^2-x_0^3,x_1^3-x_0^2x_1]$ are algebraic
varieties.}

\medskip


\item{1.28}{Let $\nu:\bP^1\to \bP^2$ be given by three homogeneous
cubic polynomials. Show that if the polynomials have no common zero,
then the image is a hypersurface which is the zero set of a cubic
polynomial.}


\medskip

\item{1.29}{Let $\nu_{\alpha,\beta}:\bP^1\to \bP^3$ be given by
$\nu_{\alpha,\beta}([x_0,x_1])=[x_0^4-\beta x_0^3x_1,x_0^3x_1-\beta
x_0^2x_1^2,
\alpha x_0^2x_1^2-x_0x_1^3,\alpha x_0x_1^3-x_1^4]$.  Show that the
image
of this map is a projective variety which is the zero locus of
one quadratic and two cubic polynomials.}



\bye
