Algebra/Topology Seminar
Speaker: David Eisenbud
Title: Weierstrass semigroups
Abstract: If X is smooth projective curve (or a compact Riemann surface) then the only globally defined regular functions on X are the constants; but if one point is removed there are many regular functions on the rest, each of which has a pole (locally 1/z^n for some n). Since the pole order of a product is the sum of the pole orders, these orders form a numerical semigroup---that is, a subset of the positive integers, closed under addition. Weierstrass proved around 1860 that this semigroup contains all but genus(X) non-negative integers; the semigroups is now called the Weierstrass semigroup of p in X, in his honor.
In 1892, Hurwitz asked: What semigroups can actually occur? The full answer is still not known! I'll explain the history of this problem and some new results, based on syzygies, from my recent work with Frank-Olaf Schreyer