Moduli of higher dimensional varieties and families of hypersurfaces

高维簇和超曲面族的模

• 批准号: 2302163
• 负责人: Kristin DeVleming
• 金额: $ 16.82万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-06-01 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302163&HistoricalAwards=false
• 关键词: Moduli; higher; dimensional; varieties; families

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.

代数几何是研究由多项式方程定义的对象，称为变量。这些方程可以用代数方法（例如，通过求解方程找到所有的解）或几何方法（例如，绘制由方程定义的形状）来研究，代数几何从这两种角度使用工具来分析变量。在这个项目中要探索的领域的一个总体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，这些空间自然地出现在代数几何、辛几何、微分几何、计数几何和组合学、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，并继续开展各种鼓励女性参与数学的活动。PI的主要目标是研究高维代数变量的模空间，特别是研究超曲面（由单个多项式方程定义的变量）的退化。PI将探讨两个相关的问题：从模理论的角度研究超曲面的光滑极限，关注这些极限何时再次成为超曲面，以及在这些模空间中出现的奇异变体的显式分类，关注射影空间的退化，Fano变体和log Calabi-Yau对。用于实现这些目标的主要工具将是k -稳定性和KSBA模的壁交叉、最小模型程序、几何不变理论以及模空间上这些不同视角之间的插值。该项目的预计输出包括几个理论结果，如射影空间的特殊退化的（不）存在性，构造log Calabi-Yau变体的模的技术，以及超曲面的光滑极限的一些分类。输出还将包括对数正则极化、对数Calabi-Yau和对数Fano对的模空间的几个显式示例。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。