Moduli spaces and maps between them

模空间和它们之间的映射

• 批准号: 1201369
• 负责人: Samuel Grushevsky
• 金额: $ 30.23万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-06-01 至 2016-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201369&HistoricalAwards=false
• 关键词: Moduli; spaces; maps; between; them

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.

PI建议研究曲线、阿贝尔簇、Prym簇和三次三倍模的几何。他将研究问题的内射映射之间的这些空间（的Torelli问题），并描述图像的这种地图（肖特基问题）。PI将进一步发展技术的亚纯微分与真实的期间，他制定了与Krichever研究几何的模空间的曲线，和奇异的平面曲线。利用他与Hulek关于三次三重的中间Jacobian的轨迹的结果，PI将试图定义一个扩展重言式环，用于阿贝尔簇的模空间的紧化，并研究其中的自然轨迹类，作为阿贝尔簇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振幅的结果来获得亏格5中经典肖特基问题的显式解。此外，PI将尝试使用他的表征，与克里切弗，普莱姆品种接近普莱姆-Torelli问题。在代数几何，一个基本问题是描述一个给定类型的所有对象的集合。给定一个代数簇（一个多项式方程组的零点集），人们可以通过变形定义方程来尝试变形它，并询问什么是变形的空间，或者询问什么是可以变形为给定的簇的空间。这些变量的参数空间被称为模空间，它们本身通常具有丰富的几何结构。此外，在许多情况下，存在与各种不同类型的各种类型相关联的构造（例如黎曼曲面的雅可比矩阵），这些构造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的信息（也就是说，图像是否决定了源--映射是否是单射的;这被称为Torelli问题），以及是否所有的变种都可以通过这样的构造得到（也就是说，映射是否是满射的;如果不是，描述图像就是肖特基问题）。拟议的项目旨在更好地理解和更明确地描述这些几何模空间的结构及其之间的关系。PI计划研究模理论中一些长期存在的开放问题，并将致力于开发研究模空间的新工具和技术。