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Failure of Boundedness for Generalised Log Canonical Surfaces

Christopher Hacon, Xiaowei Jiang

TL;DR

The paper demonstrates that boundedness for generalised log canonical (glc) models of surfaces fails in general: even with fixed invariants and ample $K_X+M$, the underlying surfaces can vary unboundedly across Kodaira dimensions. The authors construct explicit counterexamples in four geometric regimes—Calabi–Yau, Kodaira dimension one, general type, and weak Fano—where the generalized divisor $M$ has unbounded Cartier index while $K_X+M$ remains ample and its volume is fixed (e.g., $v=22$, $68$, or $45/16$ in specific constructions). They further establish boundedness results in dimension two under extra hypotheses, notably bounding the Cartier index of $M$ and handling cases where $K_X$ is big or ample, thereby delineating the precise limits of current boundedness theory for glc surfaces. Overall, the work refines our understanding of moduli and boundedness in the setting of generalized pairs and canonical bundle formulas, illustrating that the richness of surface geometry yields essential obstructions to global boundedness even in low dimension.

Abstract

In this paper, we construct counterexamples to the boundedness of generalised log canonical models of surfaces with fixed appropriate invariants, where the underlying varieties can have arbitrary Kodaira dimension. This answers a question of Birkar and the first author.

Failure of Boundedness for Generalised Log Canonical Surfaces

TL;DR

The paper demonstrates that boundedness for generalised log canonical (glc) models of surfaces fails in general: even with fixed invariants and ample , the underlying surfaces can vary unboundedly across Kodaira dimensions. The authors construct explicit counterexamples in four geometric regimes—Calabi–Yau, Kodaira dimension one, general type, and weak Fano—where the generalized divisor has unbounded Cartier index while remains ample and its volume is fixed (e.g., , , or in specific constructions). They further establish boundedness results in dimension two under extra hypotheses, notably bounding the Cartier index of and handling cases where is big or ample, thereby delineating the precise limits of current boundedness theory for glc surfaces. Overall, the work refines our understanding of moduli and boundedness in the setting of generalized pairs and canonical bundle formulas, illustrating that the richness of surface geometry yields essential obstructions to global boundedness even in low dimension.

Abstract

In this paper, we construct counterexamples to the boundedness of generalised log canonical models of surfaces with fixed appropriate invariants, where the underlying varieties can have arbitrary Kodaira dimension. This answers a question of Birkar and the first author.

Paper Structure

This paper contains 12 sections, 21 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generalised pairs and singularities
  4. Bounded family
  5. Positivity of line bundles on the blow ups of surfaces
  6. Artin's theorem on contractions
  7. Proof of Theorem \ref{['thm:bdd']}
  8. Failure of boundedness of Fglc(2,Phi,v)
  9. Calabi-Yau surfaces
  10. Minimal surfaces of Kodaira dimension one
  11. Minimal surfaces of general type
  12. Weak Fano surfaces

Key Result

Theorem 1.2

Fix $\kappa\in\{-\infty,0,1,2\}$. Let $\mathcal{P}_{\kappa}\subset\mathcal{F}_{glc}(2,\Phi,v)$ be the subset of generalised pairs such that $\kappa(X)=\kappa$ for every $(X,B+M)\in \mathcal{P}_{\kappa}$. Then $\mathcal{P}_{\kappa}$ is not bounded.

Theorems & Definitions (46)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Theorem 2.2: biancofioreHyperplaneSectionsBlowUps1989
  • Theorem 2.3: hanumanthuPositivityLineBundles2017
  • Theorem 2.4: hanumanthuPositivityLineBundles2017
  • Theorem 2.5: biancofioreHyperplaneSectionsRuled1990
  • Theorem 2.6: artinAlgebraizationFormalModuli1970
  • Theorem 3.1: birkarBoundednessVolumeGeneralised2021
  • ...and 36 more