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Local delta invariants of weak del Pezzo surfaces with the anti-canonical degree $\geq 5$

Hiroto Akaike

TL;DR

This work determines the local delta invariants $\delta_p(S)$ for all weak del Pezzo surfaces $S$ with anti-canonical self-intersection $(-K_S)^2\ge 5$, across all closed points $p\in S$. The authors implement the Abban–Zhuang framework, employing plt blowups and Zariski decompositions to reduce the computation to lower-dimensional data and to compute $S(E)$ and related quantities for each relevant divisor $E$. They achieve a complete classification for degree $5$ (seven negative-curve configurations) and degree $6$ (six configurations), providing explicit $\delta_p(S)$ values by locus (on $(-1)$-curves, $(-2)$-curves, their intersections, and outside all negative curves). The results yield precise del Pezzo surface invariants, with direct implications for the K-stability of higher-dimensional Fano varieties and applications in related stability criteria, including corollaries for smooth and du Val models.

Abstract

The delta invariant interprets the criterion for the K-(poly)stability of log terminal Fano varieties. In this paper, we determine the whole local delta invariant for all weak del Pezzo surfaces with the anti-canonical degree $\geq 5$.

Local delta invariants of weak del Pezzo surfaces with the anti-canonical degree $\geq 5$

TL;DR

This work determines the local delta invariants for all weak del Pezzo surfaces with anti-canonical self-intersection , across all closed points . The authors implement the Abban–Zhuang framework, employing plt blowups and Zariski decompositions to reduce the computation to lower-dimensional data and to compute and related quantities for each relevant divisor . They achieve a complete classification for degree (seven negative-curve configurations) and degree (six configurations), providing explicit values by locus (on -curves, -curves, their intersections, and outside all negative curves). The results yield precise del Pezzo surface invariants, with direct implications for the K-stability of higher-dimensional Fano varieties and applications in related stability criteria, including corollaries for smooth and du Val models.

Abstract

The delta invariant interprets the criterion for the K-(poly)stability of log terminal Fano varieties. In this paper, we determine the whole local delta invariant for all weak del Pezzo surfaces with the anti-canonical degree .
Paper Structure (5 sections, 24 theorems, 543 equations, 15 tables)

This paper contains 5 sections, 24 theorems, 543 equations, 15 tables.

Table of Contents

  1. Preliminaries
  2. The case of the anti-canonical degree $5$
  3. The case of the anti-canonical degree $6$
  4. The case of the anti-canonical degree $7$
  5. The case of the anti-canonical degree $8$

Key Result

Theorem 1

Let $S$ be a weak del Pezzo surface with the anti-canonical degree $5$. The symbols $(E_i, \bullet)$ and $(F_j, \circ)$ denote $(-1)$-curve and $(-2)$-curve, respectively. The local delta invariants $\delta_{p}(S)$ of $S$ at $p \in S$ are as follows. $(1)$ If the configuration of negative curves of then the local delta invariants $\delta_{p}(S)$ of $S$ at $p \in S$ are as follows. $(2)$ If the c

Theorems & Definitions (43)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 1.1: FAND,Theorem 1.106
  • Theorem 1.2: AZ,Theorem 3.2
  • Corollary 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • ...and 33 more