Precourse Worksheet 3 Homogeneous forms and rational functions on PP^1

Worksheet 3 Homogeneous forms and rational functions on PP^1

An _algebraic variety_ is a point set with locally
defined rings of polynomial functions on it. This
pronouncement covers the simplest things you can
imagine, from the affine x-line AA^1_k over
k = QQ, RR, CC, or any field you want, to the more
general foundations of scheme theory introduced by
Grothendieck in the 1960s.

Algebraic curves span 3 worlds:

Proposition. (a) An algebraic curve C in PP^n_k over
a field k has a field k(C) consisting of rational
functions on C. This function field k(C) is a
finitely generated extension of k of transcencence
degree 1. The function fields of algebraic curves
thus provide the first step beyond the Galois theory
of finite field extensions k in K.

(b) Picking a transcendental generator x displays the
function field k(C) of an algebraic curve as a finite
extension field k(x) in K, that one treats by the
methods of algebraic number theory that apply to the
number fields QQ in K. The notion of integral closure
that determines the ring of integers of a number
field also provides the resolution of singularities
of an affine algebraic curve C in AA^n_k over an
algebraically closed field k.

(c) Conversely, for k = \kbar, a function field
k in K has a model as a nonsingular projective curve
C in PP^n_k, and one proves that this model C is
unique _up to isomorphism_.

(d) A nonsingular projective algebraic curve
C in PP^n_CC over the field of complex numbers CC is
a compact Riemann surface S. The converse statement is
the fundamental theorem on the existence of global
meromorphic functions on S, for which I refer to
[Donaldson, Riemann surfaces].

The first part of my course expands on the background
from algebraic geometry, filling in several points
from the above introductory sketch.

As precourse reading and exercises, I give the elementary
treatment of the affine x-line AA^1_k and its extension
to PP^1_k, their associated coordinate ring k[x] and
homogeneous coordinate ring k[x,y], and how to use
them in simple-minded coordinate geometry. This is
mostly taken from [UAG, Chap.~1], but please bear in
mind the material around Liouville's theorems from
complex analysis.

[in preparation. This will mostly be taken from [UAG, Chap 1]