The Moduli Space of Foliated Surfaces

代数几何的区别在于，投射簇的族通常被称为模空间的单个空间自然地参数化。这个模空间的几何决定了所有变种的族的行为。特别是，曲线的模空间是数学中研究最多的种类之一。它的构造依赖于几何不变理论，但事实证明，这种方法并不能直接扩展到更高的维度。相反，一般类型的各种模空间的构造遵循Kollár和Shepherd-Barron在曲面的情况下所布置的路径，其中三维最小模型程序起着突出的作用。该建议的目的是使用最小模型程序中的最新发展来构造一个模空间，其参数化对（X，其中X是一个复射影曲面，F是X上的一般型叶状曲面，即具有大标准因子的叶状曲面。

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.