LEAPS-MPS: Describing Compactifications of Moduli Spaces of Varieties and Pairs.

LEAPS-MPS：描述簇和对模空间的紧化。

• 批准号: 2316749
• 负责人: Patricio Gallardo Candela
• 金额: $ 23.35万
• 依托单位: University of California-Riverside
• 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: 2023-09-01 至 2025-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2316749&HistoricalAwards=false
• 关键词: LEAPS; MPS; Describing; Compactifications; Moduli

Algebraic geometry is an area of mathematics that explores the geometry associated with the solution sets of polynomials. As the coefficients of these polynomials often vary, this, in turn, changes the geometry of their solution sets. Moreover, these geometric shapes carry linear data, known as their Hodge structures, which typically identify them. The PI will study the behavior of such geometric and linear data when these coefficients shift toward infinity. Particular focus is given to geometric cases that arise from applications to other mathematical areas like combinatorics and mathematical physics. Additionally, the researcher aims to foster an inclusive and diverse learning environment, providing students with training in geometric and computational tools, thereby empowering them to become scholars. This focus is particularly essential for first-generation students and underrepresented minorities, for whom the path to academia is often more uncertain.This project will study GIT, Hodge theoretic, and KSBA compactifications for the moduli space of surfaces of a general type and log pairs, such as the ones associated with configurations of plane conics and log del Pezzo surfaces with their anticanonical divisors. In particular, the PI will describe well-behaved degenerations from a Hodge theory perspective and utilize techniques such as eigenspace decompositions for studying the Limiting Mixed Hodge structures of the varieties parametrized by the relevant compact moduli spaces.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.

代数几何形状是数学领域，探索与多项式溶液集相关的几何形状。由于这些多项式的系数通常会变化，因此，这又改变了溶液集的几何形状。此外，这些几何形状具有线性数据，称为其霍奇结构，通常可以识别它们。当这些系数向无穷大时，PI将研究这种几何和线性数据的行为。特别关注从应用到组合物质和数学物理学等其他数学领域的几何案例。此外，研究人员旨在培养一个包容性和多样化的学习环境，为学生提供几何和计算工具的培训，从而赋予他们成为学者的能力。对于第一代学生和代表性不足的少数群体，这种重点尤其重要，对于他们来说，通往学术界的途径通常更不确定。本项目将研究Git，Hodge理论和KSBA压缩，以实现与一般类型和log的模量的模量空间，例如与Peezzo Surnon surnic surnon surnic surnon surnic surnic surnic surnon surnic surnic surnic surnic surnon surnon surnon surnon。特别是，PI将从霍奇理论的角度来描述行为良好的退化，并利用诸如特征空间分解等技术来研究限制的混合杂货结构，这些品种的混合杂种结构是由相关的紧凑型模态空间参数所参数的。这些奖励通过nsf的智力及其授予的智力依据，这是NSF的范围的范围。 标准。