Moduli spaces and maps between them

PI建议研究曲线的模数、阿贝尔变数、Prym变数和三次三重数的几何。他将研究这些空间之间的映射的内射性问题(Torelli问题)，以及描述这种映射的像的问题(肖特基问题)。PI将进一步发展他与Krichever共同发展的具有实周期的亚纯微分技术，以研究曲线的模空间的几何以及平面曲线的奇点。利用他和Hulek关于三次三重中间雅可比轨迹的结果，PI将尝试定义一个用于阿贝尔变种模空间紧化的扩展重言式环，并研究其中的自然轨迹类，作为阿贝尔变种退化的一种可能的归纳方法。PI还将致力于通过使用动机和他关于弦散射幅度的结果来获得亏格5的经典肖特基问题的显式解。此外，PI将尝试使用他与Krichever对Prym簇的刻画来处理Prym-Torelli问题。在代数几何中，一个基本问题是描述给定类型的所有对象的集合。给定一个代数簇(一组多项式方程的零点)，人们可以通过使定义的方程变形来尝试使它变形，并问变形的空间是什么，或者问可以变形到给定的簇的空间是多少。这些变种的参数空间被称为模空间，它们本身往往具有丰富的几何结构。此外，在许多情况下，存在与一种类型的各种、不同类型的多种相关联的结构(例如黎曼曲面的雅可比)，并且这些结构定义了一个模空间到另一个模空间的映射。人们自然会问，这些映射是否保留了所有信息(即，图像是否确定了来源--映射是否内射；这称为Torelli问题)，以及是否所有的变体都可以通过这样的构造获得(即，映射是否满射；如果不是，描述图像就是肖特基问题)。该项目的目的是更好地理解和更明确地描述这些几何模空间的结构和它们之间的关系。PI建议致力于模理论中的一些长期悬而未决的问题，并将致力于开发研究模空间的新工具和技术。

The PI proposes to study the geometry of moduli of curves, of abelian varieties, of Prym varieties, and of cubic threefolds. He will study the questions of injectivity of maps between these spaces (the Torelli problem), and of describing the images of such maps (the Schottky problem). The PI will further develop the technique of meromorphic differentials with real periods that he developed with Krichever to study the geometry of the moduli space of curves, and singularities of plane curves. Using his results with Hulek on the locus of intermediate Jacobians of cubic threefolds, the PI will attempt to define an extended tautological ring for compactifications of the moduli space of abelian varieties, and to study the classes of natural loci in it, as a possible inductive approach to degenerations of abelian varieties. The PI will also aim to obtain an explicit solution to the classical Schottky problem in genus 5, by using motivation and his results on string scattering amplitudes. Further, the PI will attempt to use his characterization, with Krichever, of Prym varieties to approach the Prym-Torelli problem.In algebraic geometry, one basic question is to describe the set of all objects of a given type. Given an algebraic variety (a zero set of a system of polynomial equations), one can try to deform it, by deforming the defining equations, and ask what is the space of deformations, or ask what is the space of varieties that can be deformed to a given one. These parameter spaces for varieties are called moduli spaces, and turn out to often have a rich geometric structure themselves. Moreover, in many instances there are constructions associating to a variety of one kind a variety of a different kind (for example the Jacobian of a Riemann surface), and these constructions define maps of one moduli space to another. It is natural to ask whether these maps preserve all the information (that is, whether the image determines the source - whether the map is injective; this is known as the Torelli problem) and whether all varieties can be obtained by such a construction (that is, whether the map is surjective; if not, describing the image is the Schottky problem). The proposed project aims to provide a better understanding and more explicit description of the structure of these geometric moduli spaces and relations among them. The PI proposes to work on some longstanding open questions in moduli theory, and will also work on developing new tools and techniques for studying moduli spaces.