Effective generation for foliated surfaces: results and applications

TL;DR

This paper investigates the birational structure and invariants of foliated surfaces using adjoint divisors, establishing bounds on automorphism groups, degrees of invariant curves, and demonstrating boundedness of certain models.

Contribution

It introduces new bounds and boundedness results for foliated surfaces of general type using adjoint divisors and invariants.

Findings

Bound on automorphism group of adjoint general type foliated surfaces.

Bound on degree of invariant curves for algebraically integrable foliations.

Set of epsilon-adjoint canonical models forms a bounded family.

Abstract

We explore the birational structure and invariants of a foliated surface $(X, \mathcal F)$ in terms of the adjoint divisor $K_{\mathcal F}+\epsilon K_X$, $0< \epsilon \ll 1$. We then establish a bound on the automorphism group of an adjoint general type foliated surface $(X, \mathcal F)$, provide a bound on the degree of a general curve invariant by an algebraically integrable foliation on a surface and prove that the set of $\epsilon$-adjoint canonical models of foliations of general type and with fixed volume form a bounded family.