David Eisenbud

ABSTRACT

If $p$ is a point on a Riemann surface (or smooth algebraic curve) $X$ then the Weierstrass semigroup of $X$ at $p$ is the set of pole orders of meromorphic (or rational) functions that are regular everywhere on $X$ except at $p$. In 1892 Hurwitz asked whether every numerical semigroup can occur in this way, but it wasn’t until 1980 that the first example of a semigroup that can not occur was found (by Ragnar Buchweitz).

I will survey what is known about Weierstrass semigroups and explain recent progress in work of mine with Frank Schreyer. We have found a new technique, using free resolutions, that has produced the simplest possible semigroup (in a precise sense) that cannot occur.