On birational boundedness of foliated surfaces

TL;DR

The paper addresses birational boundedness for foliated surfaces of general type by developing an effective pluricanonical theory in the foliated setting. It combines an analysis of Riemann–Roch for foliations with a careful study of canonical foliation singularities, including terminal, canonical non-terminal $\mathbb{Q}$-Gorenstein, and non-$\mathbb{Q}$-Gorenstein cases. A key contribution is the establishment of birational boundedness for weak nef models and, separately, for canonical models, via fixed Hilbert data $P(m)$ which fixes numerical invariants and the cusp count; this leads to uniform birational behavior of $|mK_{\mathcal{F}}|$ for large $m$. The paper also develops partial crepant resolutions to explicitly compute local RR contributions and proves strong vanishing theorems that align Hilbert functions across birational models. Together, these results advance the understanding of moduli-like questions for foliated surfaces and provide a structured pathway toward boundedness results in higher generality. The findings have significance for assessing the existence of moduli-type spaces and for controlling pluricanonical maps in foliated classification.

Abstract

In this paper we prove a result on the effective generation of pluri-canonical linear systems on foliated surfaces of general type. Fix a function $P: \mathbb Z_{\geq 0}\to \mathbb Z $, then there exists an integer $N_1>0$ such that if $(X,\mathcal F)$ is a canonical or nef model of a foliation of general type with Hilbert polynomial $χ(X, mK_{\mathcal F})=P(m)$ for all $m\in \mathbb Z_{\geq 0}$, then $|mK_{\mathcal F}|$ defines a birational map for all $m\geq N_1$. We also prove a Grauert-Riemannschneider type vanishing theorem for foliated surfaces with canonical singularities.

Theorems & Definitions (40)

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