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，称为变量。模理论的目的之一是试图通过思考一个给定的变量可能具有的所有形状的空间来理解它的性质。这个各种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空间中的变化。理解模的许多现代进展是由于退化技术-通过思考如何将给定的变量分解成更简单的对象来理解它的想法。该项目旨在进一步理解一些对代数几何至关重要的模空间的性质：三次三折的模，复曲线的模（即，从近处看像复数的物体），以及其他相关的模空间。特别是，PI将研究这些变量如何退化或分裂，并将致力于理解几何结构在这种退化下的行为。该项目将支持PI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在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构建的微分层的紧化（即具有规定的零点和极点倍数的微分曲线的模）。PI还将研究地层的双几何，并将通过应用代数几何的退化技术来解决Teichmueller动力学中的中心轨道分类问题。PI将应用实归一化微分，这是他所研究的几何学，以及他共同开发的退化技术，以实现平面曲线顶点数量的尖锐上限。同样，PI将共同研究三次三倍模空间的各种紧化的除数锥和log-MMP之间的关系。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.