Junior Geometry Seminar – Aliaksandra Novik

Abstract: Brill–Noether theory is a classical subject in algebraic geometry that studies linear series on algebraic curves. Equivalently, it studies the possible maps of a smooth curve of genus g to projective space P^r of degree d and provides the natural stratification of Pic^d(C) by the number of sections of line bundles.

The subject goes back to 1874, when Brill and M. Noether conjectured a precise numerical criterion for the existence of linear series of given degree and dimension on a general curve of genus g. In this expository talk, we will introduce the basic objects of Brill–Noether theory, state the classical Brill–Noether theorem, and sketch Lazarsfeld’s proof of it via curves on K3 surfaces. If time permits, we will also discuss a more modern approach, based on work of Bayer using Bridgeland stability conditions.