Constructions and Applications of Compactified Moduli

代数几何是研究由多项式方程定义的对象的数学领域，称为簇。模理论的目的之一是试图通过考虑给定变种可能具有的所有可能形状的空间来理解该变种的性质。这个所有形状的空间是各种给定数值类型的模空间。例如，三次三重是由一个3次方程给出的4维空间中的一种变体。在理解模数方面的许多现代进步归功于退化技术--通过思考如何将给定的变体分解成更简单的对象来理解给定的变体的想法。这个项目旨在加深对代数几何中心的一些模空间的性质的理解：三次三重的模，复曲线的模(即，从近距离看像复数的物体)，以及其他相关的模空间。特别是，PI将研究这些变体如何退化或破裂，并将致力于了解几何结构在这种退化下的行为。该项目将支持国际和平研究所和他的博士生在这些主题上的工作。该项目旨在更好地理解代数几何中普遍存在的一些模空间的紧化几何：曲线的模、带微分的曲线的模和三次三重的模。PI还将探索在Teichmueller动力学和经典代数几何问题中的应用。PI将应用他与其他研究人员共同构建的微分层(即具有指定零点和极点重数的微分的曲线的模数)的紧凑化。PI还将研究地层的双星几何，并将通过应用代数几何的退化技术来探讨Teichmueller动力学中的中心轨道分类问题。PI将应用实归一化的微分，他研究的几何，以及他共同开发的退化技术，致力于为平面曲线的尖点数量设定一个尖锐的上限。同样，PI将共同调查三次三次模空间的各种紧化的除数锥和LOG-MP之间的关系。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Algebraic geometry is the field of mathematics that studies objects defined by polynomial equations, called varieties. One of the aims of moduli theory is to try to understand properties of a given variety by thinking about the space of all possible shapes such a variety can have. This space of all shapes is the moduli space of a variety of a given numerical type. For example, the cubic threefold is a variety in 4-space given by one equation of degree 3. Much modern progress in understanding moduli is due to degeneration techniques – the idea of understanding a given variety by thinking how it can break up into simpler objects. This project aims to further the understanding of the properties of some of the moduli spaces central to algebraic geometry: the moduli of cubic threefolds, the moduli of complex curves (that is, objects that from close up look like complex numbers), and other related moduli spaces. In particular, the PI will study how such varieties can degenerate, or break up, and will aim to understand the behavior of geometric constructions under such degenerations. The project will support the work of the PI and his PhD students on these topics. The project aims to better understand the geometry of compactifications of some of the moduli spaces ubiquitous in algebraic geometry: moduli of of curves, of curves with a differential, and of cubic threefolds. The PI will also explore applications to problems in Teichmueller dynamics and in classical algebraic geometry. The PI will apply the compactification of the strata of differentials (that is, of moduli of curves with a differential with prescribed multiplicities of zeroes and poles) that he constructed with other researchers. The PI will also investigate the birational geometry of strata and will approach the central orbit classification questions in Teichmueller dynamics by applying degeneration techniques of algebraic geometry. The PI will apply real-normalized differentials, the geometry of which he studied, and the degeneration techniques he co-developed to work towards a sharp upper bound for the number of cusps of plane curves. Likewise, the PI will co-investigate the relations among cones of divisors and log-MMP for various compactifications of the moduli space of cubic threefolds.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.