The Moduli Space of Foliated Surfaces

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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.