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Minimal model program for normal pairs along log canonical locus

Kenta Hashizume

TL;DR

This work extends minimal model theory to normal pairs not necessarily lc by introducing and exploiting the relative non-nef locus $NNef(K_X+\Delta+A/Z)$ and the non-lc locus $Nlc(X,\Delta)$. It proves termination of an MMP for $(X,\Delta+A)/Z$ under the condition that these loci are disjoint, and it shows abundance for the resulting minimal model when $(K_X+\Delta+A)|_{Nlc(X,\Delta)}$ is semi-ample, leveraging quasi-log schemes induced by normal pairs to handle lc centers. The paper develops a robust framework combining relative linear systems, Nakayama–Zariski decompositions, and MMP with scaling, and it provides corollaries such as stable base locus disjointness, finite generation of section rings, and controlled MMP behavior along the lc and klt loci. The results contribute to understanding abundance and termination beyond lc pairs, with potential implications for moduli problems and birational geometry in broader singular settings.

Abstract

Let $(X,Δ)$ be a normal pair with a projective morphism $X \to Z$ and let $A$ be a relatively ample $\mathbb{R}$-divisor on $X$. We prove the termination of some minimal model program on $(X,Δ+A)/Z$ and the abundance conjecture for its minimal model under assumptions that the non-nef locus of $K_{X}+Δ+A$ over $Z$ does not intersect the non-lc locus of $(X,Δ)$ and that the restriction of $K_{X}+Δ+A$ to the non-lc locus of $(X,Δ)$ is semi-ample over $Z$.

Minimal model program for normal pairs along log canonical locus

TL;DR

This work extends minimal model theory to normal pairs not necessarily lc by introducing and exploiting the relative non-nef locus and the non-lc locus . It proves termination of an MMP for under the condition that these loci are disjoint, and it shows abundance for the resulting minimal model when is semi-ample, leveraging quasi-log schemes induced by normal pairs to handle lc centers. The paper develops a robust framework combining relative linear systems, Nakayama–Zariski decompositions, and MMP with scaling, and it provides corollaries such as stable base locus disjointness, finite generation of section rings, and controlled MMP behavior along the lc and klt loci. The results contribute to understanding abundance and termination beyond lc pairs, with potential implications for moduli problems and birational geometry in broader singular settings.

Abstract

Let be a normal pair with a projective morphism and let be a relatively ample -divisor on . We prove the termination of some minimal model program on and the abundance conjecture for its minimal model under assumptions that the non-nef locus of over does not intersect the non-lc locus of and that the restriction of to the non-lc locus of is semi-ample over .
Paper Structure (20 sections, 48 theorems, 323 equations)

This paper contains 20 sections, 48 theorems, 323 equations.

Table of Contents

  1. Introduction
  2. Minimal model theory
  3. Results
  4. Structure of this paper and idea of proofs
  5. Preliminaries
  6. $\mathbb{R}$-linear system
  7. Non-nef locus
  8. Singularities of pairs
  9. Quasi-log scheme
  10. Running minimal model program
  11. Minimal model
  12. Minimal model program
  13. Minimal model program along Kawamata log terminal locus
  14. From non-vanishing to existence of minimal model
  15. Termination of minimal model program
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $\pi \colon X \to Z$ be a projective morphism of normal quasi-projective varieties. Let $(X,\Delta)$ be a normal pair and let $A$ be a $\pi$-ample $\mathbb{R}$-divisor on $X$ such that $K_{X}+\Delta+A$ is $\pi$-pseudo-effective. Suppose that the non-nef locus of $K_{X}+\Delta+A$ over $Z$ does no Moreover, if $X$ is $\mathbb{Q}$-factorial, then all $X_{i}$ in the MMP are also $\mathbb{Q}$-facto

Theorems & Definitions (115)

  • Theorem 1.2: cf. Theorem \ref{['thm--termination-lcmmp-main']}
  • Corollary 1.3: = Corollary \ref{['cor--nonvan-lc-main']}
  • Corollary 1.4: = Corollary \ref{['cor--mmp-from-nonvan-fullgeneral']}
  • Corollary 1.5: = Corollary \ref{['cor--mmp-lc-strictnefthreshold-main']}
  • Corollary 1.6: = Corollary \ref{['cor--finite-generation-main']}
  • Definition 2.1: $\mathbb{R}$-line bundle on scheme, relative ampleness, relative semi-ampleness
  • Definition 2.2
  • Definition 2.3: Relative $\mathbb{R}$-linear system, relative stable base locus
  • Lemma 2.4: cf. bchm
  • proof
  • ...and 105 more