The Moduli Space of Foliated Surfaces

叶状曲面的模空间

• 批准号: EP/X020592/1
• 负责人: Paolo Cascini
• 金额: $ 10.28万
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX020592%2F1
• 关键词: Moduli; Space; Foliated; Surfaces

Algebraic geometry distinguishes itself in that families of projective varieties are often naturally parametrised by a single space, called a moduli space. The geometry of this moduli space determines the behaviour of families of all varieties. In particular, the moduli space of curves is one of the most heavily studied varieties in mathematics. Its construction relies on geometric invariant theory, but it turns out that this method does not extend directly to higher dimensions. Instead, the construction of moduli spaces of varieties of general type follows the path laid out by Kollár and Shepherd-Barron in the case of surfaces, where the three-dimensional Minimal Model Program plays a prominent role.The aim of this proposal is to use the recent developments in the Minimal Model Program for foliations over a threefold to construct a moduli space which parametrises pairs (X,F) where X is a complex projective surface and F is a foliation on X of general type, i.e. with big canonical divisor, under some natural assumptions on their singularities.

代数几何区别本身本身，因为射影品种的家族通常是由单个空间（称为模量空间）自然参数参数的。该调节空间的几何形状决定了所有品种家族的行为。尤其是，曲线的模量空间是数学中最重的研究品种之一。它的构造依赖于几何不变理论，但事实证明，该方法并未直接扩展到更高的维度。取而代之的是，在表面的情况下，Kollár和Shepherd-Barron所规定的一般类型变化空间的构建遵循了三维最小模型计划的重要作用。该提议的目的是在三维范围内扮演的目的。表面和f是一般类型的X叶叶，即具有大规范除数，在其奇异性的一些自然假设下。