Prerequisites (may be edited)

The course uses many items from u/g algebra:
  linear algebra, quadratic forms.
from Galois theory:
  field extensions, the distinction between algebraic and transcendental
  extension. Algebraically closed fields. The distinction between separable
  and inseparable extension.
  (Only very basic treatments are needed; some of it will be expanded
  during the course, or stated with proofs in appendix.)
from Commutative algebra:
  commutative rings, ideals, modules. Fields of fractions, UFD.
  prime and maximal ideals. Noetherian rings and modules.
from Complex analysis (mostly for motivation):
  Meromorphic functions, their zeros and poles. The principal part
  at a pole. The Cauchy integral formula.
from Algebraic geometry (all of this will be recalled briefly):
  algebraic set V(I), the coordinate ring k[V] of an affine variety.
  projective variety V in projecctive space PP^n and its field of
  fractions.

Comments on the prerequisites: an algebraic curve C over an
algebraically closed field k has a function field k(C), that is a field
extension k in k(C) of transcendence degree 1. So k in k(C) is not an
algebraic extension, but the next best thing. Acccording to the
student's individual needs, we could assume the k = CC throughout, or k
of characteristic 0, in which case we could ignore separable. The
student coming from number theory will have some general ideas about
what to do if the field is not algebraically closed.

The course uses basic ideas from commutative algebra throughout, and
these will be explained and developed. The local ring of a curve at a
point P is a local ring of Krull dimension 1 (that is, has exactly 2
prime ideals, 0 and the maximal ideal of functions vanishing at P; any
function not zero at P is invertible in the local ring). At a
nonsingular point, the local ring is a DVR (discrete valuation ring),
which is the simplest kind of UFD (unique factorisation domain), with
only a single prime element. An affine variety V has a coordinate ring
k[V], and for a curve, the notion of nonsingularity and resolution of
singularities are treated in terms of integral closure and normalisation
of the affine coordinate ring.