Moduli of higher dimensional varieties and families of hypersurfaces

代数几何是研究由多项式方程定义的对象，称为品种。 这些方程可以用代数学（例如求解方程以找到所有解）或几何学（例如绘制方程定义的形状）来研究，代数几何学从两个角度使用工具来分析各种各样的问题。 在这个项目中要探索的领域的首要目标是对所有可能的品种进行分类，这是通过构建模空间或给定类型的品种的参数空间来完成的。 模空间的研究有着丰富的历史，这些空间自然出现在代数几何、辛几何、微分几何、枚举几何和组合数学、镜像对称、数论和物理学中。 这个项目所涉及的工作与这些领域都有联系。PI将指导本科生和研究生，并继续开展各种活动，鼓励妇女参与数学，PI的主要目标是研究高维代数簇的模空间，特别是研究超曲面（由单一多项式方程定义的簇）的退化。 PI将处理两个相关的问题：从模论的角度研究超曲面的光滑极限，重点是当这种极限再次超曲面，也是一个明确的分类奇异品种出现在这些模空间，专注于退化的射影空间，法诺品种，和日志卡-丘对。 用于实现这些目标的主要工具将是K-稳定性和KSBA模中的跨壁，最小模型程序，几何不变理论以及模空间上这些不同观点之间的插值。 该项目的预计产出包括几个理论结果，如（不）存在的特殊退化的射影空间，技术构造模的日志卡-丘品种，和一些分类的光滑极限的超曲面。 输出还将包括对数正则极化、对数卡拉比-丘和对数范诺对的模量空间的几个明确例子。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

Algebraic geometry is the study of objects defined by polynomial equations, called varieties. These equations can be studied algebraically (e.g. solving the equation to find all solutions) or geometrically (e.g. graphing the shape defined by the equation) and algebraic geometry uses the tools from both perspectives to analyze varieties. An overarching goal of the field to be explored in this project is to classify all possible varieties, which is done through the construction of moduli spaces, or parameter spaces for varieties of a given type. The study of moduli spaces has a rich history and these spaces arise naturally in algebraic geometry, symplectic geometry, differential geometry, enumerative geometry and combinatorics, mirror symmetry, number theory, and physics. The work involved with this project has connections to each of these fields. The PI will mentor both undergraduate and graduate students and continue with a variety of activities that encourage the participation of women in mathematics.The main objective of the PI is to research moduli spaces of higher dimensional algebraic varieties, specifically to study degenerations of hypersurfaces (varieties defined by a single polynomial equation). The PI will approach two related questions: studying smooth limits of hypersurfaces from a moduli-theoretic perspective, focusing on when such limits are again hypersurfaces, and also an explicit classification of singular varieties appearing in these moduli spaces, focusing on degenerations of projective space, Fano varieties, and log Calabi-Yau pairs. The main tools used to accomplish these goals will be wall crossing in K-stability and KSBA moduli, the minimal model program, geometric invariant theory, and interpolation between these different perspectives on moduli spaces. The projected outputs of this project include several theoretical results, such as (non-)existence of particular degenerations of projective space, techniques for constructing moduli of log Calabi-Yau varieties, and some classification of smooth limits of hypersurfaces. The outputs will also include several explicit examples of moduli spaces of log canonically polarized, log Calabi-Yau, and log Fano pairs.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.