Canonical Models of Adjoint Foliated Structures on Surfaces

TL;DR

This paper explores the structure and classification of adjoint foliated surfaces, focusing on their minimal models and the behavior of associated linear systems, providing solutions to boundedness problems in algebraic geometry.

Contribution

It introduces a detailed study of canonical models of adjoint foliated structures on surfaces and addresses a boundedness problem for foliated surfaces of general type.

Findings

Effective description of linear systems |m(K_F + D)| for large m

Resolution of a boundedness problem for foliated surfaces of general type

Development of canonical models for adjoint foliated structures

Abstract

In this paper, we study the adjoint foliated structures of the form $K_{\mathcal{F}}+D$ on algebraic surfaces, with particular focus on their minimal and canonical models. We investigate the effective behavior of the multiple linear system $|m(K_{\mathcal{F}}+D)|$ for sufficiently divisible integers $m>0$. As an application, we provide an effective answer to a boundedness problem for foliated surfaces of general type, originally posed by Hacon and Langer.