The Moduli Space of Foliated Surfaces

代数几何的独特之处在于，射影变族通常被一个称为模空间的空间自然参数化。这个模空间的几何形状决定了所有变种族的行为。特别是曲线的模空间是数学中研究得最多的一类。它的构造依赖于几何不变理论，但事实证明，这种方法不能直接推广到高维。相反，一般类型的各种模空间的构造遵循Kollár和Shepherd-Barron在曲面情况下布置的路径，其中三维最小模型程序起着突出的作用。本文的目的是利用三曲面上叶形的最小模型规划的最新进展，在其奇异性的一些自然假设下，构造一个参数化对（X，F）的模空间，其中X是一个复射影曲面，F是X上的一般型叶形，即具有大正则除数。

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.