Syzygies and moduli

该提案的所有项目的显著特点是对合相的变分方法。有两个主要的目标集：一方面，找到一些悬而未决的问题的答案syzygies代数曲线：的Prym-Green猜想paracanonical曲线的甚至属;描述一套可能的决议规范曲线的固定gonality的精神Schreyer的猜想。 另一方面，我们用合冲方法探讨了模理论中的一些主要问题。通过研究曲线在其标准丛的中合点处的映射，刻画了大亏格曲线的模空间M_g的标准模型。我们希望确定的双有理几何的Hurwitz空间的覆盖的投影线的固定度和亏格的渐近特征。这两组问题密切相关。模问题的进展，预计将实现通过新的几何相关的通用syzygy建设上的模堆栈的问题。相反，一个单独的合冲问题可以被变分处理，这样它就变成了一个模栈上的横截性陈述，可以用枚举、退化、几何不变论、Hodge理论或导出范畴方法来处理。

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.