Syzygies and moduli

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The salient feature characterizing all the projects of the proposal is a variational approach to syzygies. There are two major sets of goals : On one side, find an answer to some of the outstanding questions on syzygies of algebraic curves: The Prym-Green Conjecture for paracanonical curves of even genus; describe the set of possible resolutions of canonical curves of fixed gonality in the spirit of Schreyer's Conjecture. On the other side, we pursue with syzygetic methods major questions in moduli theory. We aim to describe the canonical model of the moduli space M_g of curves of large genus g, by studying the map associated to a curve the middle syzygy point of its canonical bundle. We wish to determine the asymptotic features of the birational geometry of the Hurwitz space of covers of the projective line of fixed degree and genus. The two sets of questions are intimately related. Progress on moduli questions is expected to be achieved via new geometries associated to universal syzygy construction on the moduli stack on question. Conversely, an individual syzygetic question can be treated variationally, so that it becomes a transversality statement on a moduli stack, which can be treated with enumerative, degeneration, Geometric Invariant Theory, Hodge-theoretic or derived category methods.