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$δ$-invariants of log Fano planes

Elena Denisova

TL;DR

This work determines the δ-invariant for pairs $(\mathbb{P}^2, \lambda C_d)$ with $d\le 4$, providing explicit stability windows for a wide range of singularity types and tangency configurations. The authors employ Kento Fujita’s formulas for the δ-invariant and leverage the Abban–Zhuang method to reduce higher-dimensional stability questions to surface computations on plane curves, yielding precise expressions and bounds for $\delta_P(\mathbb{P}^2,\lambda C)$ and $\delta(\mathbb{P}^2,\lambda C)$. As a result, they produce new examples of $K$-stable and $K$-semistable log Fano pairs, with applications to Du Val del Pezzo surfaces, projective spaces, cubic and quartic solids, and complete intersections, thereby enriching the $K$-moduli landscape. The findings illuminate how singularity type and tangency influence stability, and demonstrate a practical framework for examining $K$-stability in low-dimensional settings through systematic surface-level analysis.

Abstract

We compute the $δ$-invariant for pairs $(\mathbb{P}^2, λC_d)$, where $C_d$ is a plane curve of degree $d \leq 4$. These computations provide new examples of $K$-stable and $K$-semistable log Fano pairs, and contribute to the study of $K$-stability of log Fano varieties via the Abban-Zhuang method, which reduces higher-dimensional problems to the surface case.

$δ$-invariants of log Fano planes

TL;DR

This work determines the δ-invariant for pairs with , providing explicit stability windows for a wide range of singularity types and tangency configurations. The authors employ Kento Fujita’s formulas for the δ-invariant and leverage the Abban–Zhuang method to reduce higher-dimensional stability questions to surface computations on plane curves, yielding precise expressions and bounds for and . As a result, they produce new examples of -stable and -semistable log Fano pairs, with applications to Du Val del Pezzo surfaces, projective spaces, cubic and quartic solids, and complete intersections, thereby enriching the -moduli landscape. The findings illuminate how singularity type and tangency influence stability, and demonstrate a practical framework for examining -stability in low-dimensional settings through systematic surface-level analysis.

Abstract

We compute the -invariant for pairs , where is a plane curve of degree . These computations provide new examples of -stable and -semistable log Fano pairs, and contribute to the study of -stability of log Fano varieties via the Abban-Zhuang method, which reduces higher-dimensional problems to the surface case.

Paper Structure

This paper contains 42 sections, 88 theorems, 385 equations, 9 figures.

Table of Contents

  1. Introduction
  2. History and Results
  3. Applications.
  4. Du Val del Pezzo surfaces of degree $2$
  5. Projective Space
  6. Cubic Threefolds
  7. Quartic Double Solids
  8. Sextic Double Solids
  9. Complete intersection of a quadric and a cubic in $\mathbb{P}^5$
  10. Proof of Main Theorem via Kento Fujita’s formulas
  11. Line
  12. Smooth conic
  13. Smooth cubic curves
  14. Smooth quartic curves
  15. Curves with $A_1$ singularities
  16. ...and 27 more sections

Key Result

Theorem 1.3

Let $X$ be a $\mathbb{Q}$-Fano variety. If $\alpha(X) > \frac{n}{n+1}$, then $X$ is $K$-stable, where

Figures (9)

  • Figure 1: Cubic Curves
  • Figure 2: Irreducible Quartic Curves
  • Figure 3: Line and Irreducible cubic
  • Figure 4: Two Conics
  • Figure 5: Conic and Two Lines
  • ...and 4 more figures

Theorems & Definitions (143)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3: Tian's criterion
  • Remark 1.4
  • Lemma 1.5
  • proof
  • Lemma 1.6
  • proof
  • Corollary 1.7
  • Corollary 1.8
  • ...and 133 more