Frank Schreyer's Lectures at Oberwolfach
Lecture 1:  Groebner Basics

-> Ideal Membership Problem
   * Easy in 1-variable
   * Easy to get messed up trying in 2 (or more) variables
   * Notion of monom order

-> Monomial orders
   * Lexicographic, real weights, degree reverse lexicographic
   * A monom order is artinian  (uses Dixon's Lemma)
   * Leading term of a polynomial

-> Division with remainder of a polynomialk by a list.
   * Its dependancy on the order of the f_i

-> Definition of a Groebner basis in terms of the LT(ideal)
   * Proposition.  Division algorithm works for Groebner bases,
      "A Groebner basis provides a solution to ideal membeship problem"
   * Gordan's proof of the Hilbert Basis Theorem

-> Detection of Groebner bases
   * Definition of S-pairs (motivated for detection of new monomials in Lt(I)).
   * Buchberger's Criterion
     -- proof uses free module Groebner bases, and begins to give information 
        about syzygies

-> Definition of (first) syzygy module
    A finitely generated k[x1,..,xn] module has a finite free resolution
    
-> Application: Hilbert function is a polynomial (eventually)

Exercise:  Let P^d x P^e ---> P^{de+d+e} be the Segre embedding
           (xi) x (yj) |---> (zij = xi*yj)
           Exercise:  Show the 2x2 minors of the matrix (zij) are a GB.

        For Small d and e, compute the Hilbert resolution (by hand)

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Lecture 2:

W^r_d(C) = set of line bundles of degree d on C with at least r+1 sections
           (Subset of Jacobian)
Brill-Noether Theory implies that its dimension is at least
           g-(r+1)(g-d+r) = rho(g, r, d)
     For a general curve this is an equality and W^r_d - W^{r+1}_d is smooth.

Idea is to use W^r_d to dominate M_g for rho as big as possible

g<= 10, a plane curve model works.  Can find a family linear series over a
          family of plane curves, giving a unirational parametrization of M_g

--> This fails for g=11.  We try space curves. (Degree 12, genus 11)
    There could be a curve of this degree with 6 generators of its ideal
    in degree 5.  Finding one such curve (with the right ranks of maps between
    higher cohomology of C and its ideal sheaf) will imply the general one has
    these data, and then that this rational family dominates M_11.