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Canonical Models of Adjoint Foliated Structures on Surfaces

Jun Lu, Xiaohang Wu, Shi Xu

TL;DR

The paper develops a comprehensive framework for adjoint foliated structures $K_{\mathcal{F}}+D$ on algebraic surfaces, focusing on minimal and canonical models and the pseudo-effective regime. By introducing $(D,\mathcal{F})$-chains and proving that the negative part in the Zariski decomposition is the disjoint union of maximal such chains, it obtains a precise structural description of $K_{\mathcal{F}}+D$ and its canonical model. Leveraging Tan’s results on the effective behavior of linear systems, the authors derive explicit bounds ensuring basepoint-freeness and very ampleness for adjoint divisors on the canonical model, and establish boundedness results for foliated surfaces of general type. The work yields an effective resolution to a boundedness problem posed by Hacon and Langer and provides a robust toolkit for studying adjoint foliated structures through explicit combinatorial and geometric data.

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.

Canonical Models of Adjoint Foliated Structures on Surfaces

TL;DR

The paper develops a comprehensive framework for adjoint foliated structures on algebraic surfaces, focusing on minimal and canonical models and the pseudo-effective regime. By introducing -chains and proving that the negative part in the Zariski decomposition is the disjoint union of maximal such chains, it obtains a precise structural description of and its canonical model. Leveraging Tan’s results on the effective behavior of linear systems, the authors derive explicit bounds ensuring basepoint-freeness and very ampleness for adjoint divisors on the canonical model, and establish boundedness results for foliated surfaces of general type. The work yields an effective resolution to a boundedness problem posed by Hacon and Langer and provides a robust toolkit for studying adjoint foliated structures through explicit combinatorial and geometric data.

Abstract

In this paper, we study the adjoint foliated structures of the form on algebraic surfaces, with particular focus on their minimal and canonical models. We investigate the effective behavior of the multiple linear system for sufficiently divisible integers . As an application, we provide an effective answer to a boundedness problem for foliated surfaces of general type, originally posed by Hacon and Langer.
Paper Structure (19 sections, 53 theorems, 142 equations, 1 figure)

This paper contains 19 sections, 53 theorems, 142 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Foliations and singularities
  4. Index Theorems
  5. Non-invariant curves
  6. Invariant curves
  7. Technical tools for algebraic surfaces
  8. (D,F)-chains
  9. F-chains
  10. (D,F)-chains
  11. Curves outside (D,F)-chains
  12. Zariski decomposition of KF+D
  13. Proof of Theorem \ref{['mainthm:<1/4']}
  14. Other Variants
  15. Canonical model of KF+D
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $(X,\mathcal{F})$ be a log minimal foliated surface (cf. Definition def:logminfoliation). Assume $\epsilon\in(0,\frac{1}{4})\cap\mathbb{Q}$ and that $K_{\mathcal{F}}+\epsilon K_X$ is pseudo-effective. Let $K_{\mathcal{F}}+\epsilon K_X = P(\epsilon)+N(\epsilon)$ be its Zariski decomposition. Then

Theorems & Definitions (110)

  • Theorem 1.1: = Theorem \ref{['mainthm:<1/4']} + Theorem \ref{['thm:Null']} + Corollary \ref{['coro:canmod0<e<1/4']}
  • Corollary 1.2: = Theorem \ref{['them:boundednessofD']} + Corollary \ref{['coro:boundednessofeKX']}
  • Remark 1.3
  • Theorem 1.4: = Theorem \ref{['mainthm:<1/4']} + Theorem \ref{['thm:Null']} + Corollary \ref{['coro:canmod-e=0']}
  • Remark 1.5
  • Proposition 1.6: = Theorem \ref{['them:boundednessofD']} + Proposition \ref{['prop:P(D)H1']}
  • Corollary 1.7
  • Remark 1.8
  • Proposition 1.9: = Corollary \ref{['cor:eKXcanonical']}
  • Definition 2.1
  • ...and 100 more