On moduli of foliated surfaces

Abstract

The birational geometry of foliations exhibits phenomena that are fundamentally different than the case of varieties and that complicate the construction of a satisfactory moduli theory: the canonical class of a minimal foliation of general type need not determine a polarization; canonical singularities are not stable under deformation; and a family of foliations may degenerate to a non-saturated integrable distribution. These features imply that a new notion of stability and new techniques are required to control degenerations. In this paper, we develop a theory of stable foliated surfaces and construct their moduli spaces. To achieve these goals, we introduce semi-log canonical foliations and identify a class of stable integrable distributions on surfaces which is preserved under degeneration. We also establish the corresponding boundedness, deformation, separatedness, and stable-reduction results. Some of the main technical ingredients are inversion of adjunction and deformation-openness for log canonical singularities of rank-one foliations. Using these new results, we show that stable foliated surfaces with fixed Cartier index and adjoint volume form a Deligne--Mumford stack of finite type over $\mathbb{C}$. Allowing the numerical invariants to vary yields a Deligne-Mumford stack locally of finite type which satisfies the valuative criterion for properness for families with normal generic fibre. This provides a compactification framework for moduli of surface foliations and links their birational classification with their deformation theory.