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On K-stability of singular hyperelliptic Fano 3-folds

Hamid Abban, Ivan Cheltsov, Adrien Dubouloz, Kento Fujita, Takashi Kishimoto, Jihun Park

TL;DR

The paper classifies the K-stability of singular hyperelliptic Fano 3-folds across 47 deformation families by leveraging the anti-log-canonical model and delta-invariants, together with toric degeneration techniques from Fujita2024 and Abban–Zhuang theory, supplemented by α-invariant and equivariant methods. It proves that general members of H_5,H_7,H_8,H_{11},H_{12},H_{13} are K-stable, identifies unique K-polystable members in H_{10} and H_{17}, and shows all other families H_{14}–H_{47} are K-unstable, with a corollary on ample anticanonical divisors for large degree. The results are obtained through a unified framework that combines toric calculations, plt-flags, and equivariant stability criteria, illustrating how modern K-stability techniques apply to singular hyperelliptic Fano 3-folds. These findings provide a near-complete stability map for this class and demonstrate the practical power of the Fujita2024–Abban–Zhuang approach in the explicit classification of K-stability.

Abstract

We study the K-stability of singular Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system is base-point-free but not very ample.

On K-stability of singular hyperelliptic Fano 3-folds

TL;DR

The paper classifies the K-stability of singular hyperelliptic Fano 3-folds across 47 deformation families by leveraging the anti-log-canonical model and delta-invariants, together with toric degeneration techniques from Fujita2024 and Abban–Zhuang theory, supplemented by α-invariant and equivariant methods. It proves that general members of H_5,H_7,H_8,H_{11},H_{12},H_{13} are K-stable, identifies unique K-polystable members in H_{10} and H_{17}, and shows all other families H_{14}–H_{47} are K-unstable, with a corollary on ample anticanonical divisors for large degree. The results are obtained through a unified framework that combines toric calculations, plt-flags, and equivariant stability criteria, illustrating how modern K-stability techniques apply to singular hyperelliptic Fano 3-folds. These findings provide a near-complete stability map for this class and demonstrate the practical power of the Fujita2024–Abban–Zhuang approach in the explicit classification of K-stability.

Abstract

We study the K-stability of singular Fano 3-folds with canonical Gorenstein singularities whose anticanonical linear system is base-point-free but not very ample.
Paper Structure (8 sections, 11 theorems, 79 equations)

This paper contains 8 sections, 11 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Proof of the Main Theorem
  3. K-unstable cases
  4. Alternative Approach for K-unstable cases
  5. K-stable cases
  6. Alternative Approach for K-stable cases
  7. K-polystable cases
  8. Alternative Approach for K-polystable cases

Key Result

Corollary 1.5

Let $X$ be a Fano 3-fold with canonical Gorenstein singularities and $(-K_X)^3\geqslant 14$. Suppose that $X$ is K-semistable. Then $-K_X$ is very ample.

Theorems & Definitions (25)

  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • ...and 15 more