On birational boundedness of foliated surfaces

TL;DR

This paper establishes effective bounds for the birationality of pluri-canonical systems on foliated surfaces of general type and proves a vanishing theorem for such surfaces with canonical singularities.

Contribution

It provides a uniform bound ensuring the birationality of pluricanonical maps for foliated surfaces of general type, extending classical results to foliated contexts.

Findings

Existence of a universal integer N_1 for birationality of |mK_F|

Effective generation of pluri-canonical linear systems

A Grauert-Riemenschneider type vanishing theorem for foliated surfaces

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 $\chi (X, mK_{\mathcal F})=P(m)$ 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.