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Optimal bounds for local volumes of threefold singularities

Yuchen Liu

TL;DR

This work proves an optimal bound for local volumes of Gorenstein canonical non-hypersurface threefold singularities, showing $\widehat{vol}(x,X)\le 9$ with equality only for $\frac{1}{3}(1,1,1)$ quotients. The authors blend hypersurface monomial-valuation bounds, crepant-exceptional-divisor analysis, Artin-approximation-based factorization, and MMP techniques to handle all crepancy cases, culminating in a dimension-3 analogue of a refined ODP-gap-type result. The results yield new restrictions on singularities in K-moduli spaces of Fano threefolds and establish a sharp inequality relating local volumes to minimal log discrepancies, enhancing the understanding of local-to-global behavior in threefold singularities. Overall, the paper provides a comprehensive framework for bounding local volumes in dimension three and for applying these bounds to moduli problems and singularity theory.

Abstract

We establish an optimal upper bound for local volumes of Gorenstein canonical non-hypersurface threefold singularities. Specifically, we show that a klt threefold singularity with local volume at least $9$ is either a hypersurface singularity or a quotient singularity. As applications, we obtain new restrictions on the singularities of members in K-moduli spaces of Fano threefolds, and we establish a sharp inequality between local volumes and minimal log discrepancies for threefold singularities.

Optimal bounds for local volumes of threefold singularities

TL;DR

This work proves an optimal bound for local volumes of Gorenstein canonical non-hypersurface threefold singularities, showing with equality only for quotients. The authors blend hypersurface monomial-valuation bounds, crepant-exceptional-divisor analysis, Artin-approximation-based factorization, and MMP techniques to handle all crepancy cases, culminating in a dimension-3 analogue of a refined ODP-gap-type result. The results yield new restrictions on singularities in K-moduli spaces of Fano threefolds and establish a sharp inequality relating local volumes to minimal log discrepancies, enhancing the understanding of local-to-global behavior in threefold singularities. Overall, the paper provides a comprehensive framework for bounding local volumes in dimension three and for applying these bounds to moduli problems and singularity theory.

Abstract

We establish an optimal upper bound for local volumes of Gorenstein canonical non-hypersurface threefold singularities. Specifically, we show that a klt threefold singularity with local volume at least is either a hypersurface singularity or a quotient singularity. As applications, we obtain new restrictions on the singularities of members in K-moduli spaces of Fano threefolds, and we establish a sharp inequality between local volumes and minimal log discrepancies for threefold singularities.

Paper Structure

This paper contains 13 sections, 28 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local volumes
  4. Threefold hypersurface singularities
  5. Crepant exceptional divisors
  6. Gorenstein elliptic surface singularities
  7. Factorizing contractions by Artin approximation
  8. Proofs
  9. Hypersurface singularities
  10. Cases when $\mathrm{crex}=1$
  11. Cases when $\mathrm{crex} = 2$
  12. General cases
  13. Applications and examples

Key Result

Theorem 1.1

Let $x\in X$ be a Gorenstein canonical non-hypersurface threefold singularity. Then $\widehat{\mathrm{vol}}(x,X)\leq 9$, and equality holds if and only if $x\in X$ is a quotient singularity of type $\frac{1}{3}(1,1,1)$.

Theorems & Definitions (60)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: Stable Degeneration Theorem, Blu18LX18LWX18XZ20BLQ24XZ25
  • Definition 2.4
  • Example 2.5: LL19LX20
  • Theorem 2.6: XZ20
  • ...and 50 more