Robert Lazarsfeld
University of Michigan, Ann Arbor

Title: Asymptotic syzygies of algebraic varieties

Abstract: I'll present joint work with Lawrence Ein concerning the asymptotic behavior of the syzygies of a smooth projective variety X as the positivity of the embedding line bundle grows. We prove that at least as far as grading is concerned, the minimal resolution of the ideal of X has a surprisingly uniform asymptotic shape: roughly speaking, generators eventually appear in almost all degrees permitted by Castelnuovo-Mumford regularity. For Veronese embeddings of projective space, we give an effective statement that in some cases is optimal, and conjecturally always is so. Finally, I will discuss some work in progress (with Ein, Erman and Lee) concerning the asymptotics of the betti numbers of the resolutions in question.