Numerical conditions for the boundedness of foliated surfaces

TL;DR

This paper establishes finiteness results for Hilbert functions of 2D foliated canonical models under fixed numerical invariants, impacting the understanding of their boundedness and effective birationality.

Contribution

It proves finiteness of Hilbert functions for foliated surfaces with fixed invariants and extends known results on boundedness and birationality under weaker conditions.

Findings

Finiteness of Hilbert functions for fixed invariants.

Extension of boundedness results to broader conditions.

Examples illustrating properties of canonical model families.

Abstract

We show that the set of Hilbert functions $P(m)=\chi(mK_\mathcal{F})$ of 2-dimensional foliated canonical models with fixed $K_\mathcal{F}^2$, $K_\mathcal{F} \cdot K_X$ and $i_\mathbb{Q}(\mathcal{F})$ is finite. As a consequence, we deduce that two results on the effective birationality and boundedness of foliated canonical models with fixed Hilbert function still hold when only $K_\mathcal{F}^2$, $K_\mathcal{F} \cdot K_X$ and $i_\mathbb{Q}(\mathcal{F})$ are fixed. We then give examples further investigating the properties of families of canonical models, and study particular cases in which some of the conditions are not necessary.