Syzygies and moduli

Syzygies 和模数

• 批准号: 388129650
• 负责人: Professor Dr. Gavril Farkas
• 金额: --
• 依托单位: Department of Mathematics
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2017
• 资助国家: 德国
• 起止时间: 2016-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/388129650?language=en
• 关键词: Syzygies; moduli

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.

该提案的所有项目的显著特点是采用变分方法进行融合。这里有两个主要的目标：一方面，找到代数曲线合集的一些悬而未决的问题的答案：偶亏格的副正则曲线的Prym-Green猜想；在Schreyer猜想的精神下，描述固定正则性的标准曲线的可能的分解集。另一方面，我们用合谋的方法探讨了模理论中的主要问题。通过研究与曲线的标准丛的中间合点相关的映射，刻画了大亏格g的曲线的模空间M_g的标准模型。我们想要确定定次亏格射影直线的覆盖的Hurwitz空间的双调几何的渐近特征。这两组问题是密切相关的。模问题的进展有望通过与所讨论的模堆叠上的泛合构造相关联的新几何来实现。反之，单个合取问题可以变分处理，从而成为模栈上的横断性陈述，可以用计数、退化、几何不变理论、Hodge理论或派生范畴方法来处理。