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Del Pezzo surfaces with four log terminal singularities

Grigory Belousov, DongSeon Hwang

TL;DR

This work completes the classification of log del Pezzo surfaces of Picard number one carrying exactly four log terminal singularities by leveraging $ abla$P$^1$-fibration structures on the minimal resolution, detailed dual-graph (Hirzebruch–Jung) analysis, and a suite of MMP/BMY–type arguments. The authors dissect the problem into cases according to the nonemptiness of $|C+D+K_{ar X}|$ and whether singularities are cyclic or non-cyclic, performing exhaustive, geometry-first casework that combines contraction sequences, Hurwitz-type formulas, and fibre analysis. The main result lists realizable singularity configurations and demonstrates that many prospective patterns are ruled out, yielding a precise, graph-theoretic description of all valid four-point configurations. This advances the understanding of the boundary cases in the moduli of del Pezzo surfaces and provides explicit combinatorial data for further geometric and arithmetic applications.

Abstract

We classify del Pezzo surfaces with Picard number is equal to one and with four log terminal singular points.

Del Pezzo surfaces with four log terminal singularities

TL;DR

This work completes the classification of log del Pezzo surfaces of Picard number one carrying exactly four log terminal singularities by leveraging P-fibration structures on the minimal resolution, detailed dual-graph (Hirzebruch–Jung) analysis, and a suite of MMP/BMY–type arguments. The authors dissect the problem into cases according to the nonemptiness of and whether singularities are cyclic or non-cyclic, performing exhaustive, geometry-first casework that combines contraction sequences, Hurwitz-type formulas, and fibre analysis. The main result lists realizable singularity configurations and demonstrates that many prospective patterns are ruled out, yielding a precise, graph-theoretic description of all valid four-point configurations. This advances the understanding of the boundary cases in the moduli of del Pezzo surfaces and provides explicit combinatorial data for further geometric and arithmetic applications.

Abstract

We classify del Pezzo surfaces with Picard number is equal to one and with four log terminal singular points.

Paper Structure

This paper contains 16 sections, 34 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some useful theorems in algebraic geometry
  4. Log del Pezzo surfaces of Picard number one
  5. $\mathbb{P}^1$-fibration structures on $\bar{X}$
  6. The case $|C+D+K_{\bar{X}}|\neq\emptyset$
  7. The case $|C+D+K_{\bar{X}}|=\emptyset$
  8. The case $(2,3,5)$
  9. The case $(2,3,4)$
  10. The case $(2,3,3)$
  11. The Case with a non-cyclic singular point that is not a rational double point
  12. $Z=\mathbb{P}^1$
  13. $Z$ is a surface.
  14. Assume that $E$ passes through two singular points of $X$: one on $D_1$ and the other outside $D_1$.
  15. Assume that $E$ passes through two singular points of $X$ outside $D_1$.
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $X$ be a log del Pezzo surface of Picard number one. Assume that $X$ has four singular points. Then the singularity type of $X$ is one of the following, and they are all realizable:

Theorems & Definitions (59)

  • Theorem 1.1
  • Theorem 2.1: Hurwitz, see, e.g., Har, Corollary 2.4, Ch. 4
  • Theorem 2.2: Hodge, see, e.g., Har, Theorem 1.9, Remark 1.9.1, Ch. 5
  • Theorem 2.3: see KeM
  • Theorem 2.5: Hw
  • Lemma 2.6: Zh
  • Lemma 2.7
  • proof
  • Definition 2.8
  • Lemma 2.9: Zh
  • ...and 49 more