# Syzygies, gonality and symmetric products of curves

 IAS/Princeton Algebraic Geometry Day Topic: Syzygies, gonality and symmetric products of curves Speaker: Robert Lazarsfeld Affiliation: Stony Brook University Date: Tuesday, April 14 Time/Room: 4:00pm - 5:00pm/S-101 Video Link: http://video.ias.edu/puias/2015/0414-RobertLazarsfeld

In the mid 1980s, Mark Green and I conjectured that one could read off the gonality of an algebraic curve $C$ from the syzygies among the equations defining any one sufficiently positive embedding of $C$. Ein and I recently noticed that a small variant of the ideas used by Voisin in her work on canonical curves leads to a quick proof of this gonality conjecture. The proof involves the geometry of certain vector bundles on the symmetric product of $C$. I will describe this circle of ideas, and I will also discuss a partial generalization, with Ein and Yang, to smooth varieties of all dimensions.