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Bounding geometrically integral del Pezzo surfaces

Fabio Bernasconi, Gebhard Martin

TL;DR

This paper advances the understanding of boundedness for del Pezzo surfaces over arbitrary fields by proving sharp irregularity bounds for geometrically integral normal locally complete intersection del Pezzo surfaces and showing wild canonical del Pezzo surfaces occur only in characteristic $2$. It extends Tanaka’s boundedness results to the canonical case, and establishes BAB-type boundedness for geometrically integral canonical and certain $ ext{ε}$-klt del Pezzo surfaces in characteristic not equal to $2,3,5$, using a combination of Frobenius techniques, $ ext{α}$-torsors, and effective vanishing. Key contributions include effective vanishing and very ampleness results for anti-pluricanonical systems, a detailed analysis of irregularity via the Frobenius length and δ-invariants, and explicit volume and index bounds that lead to bounded families over $ ext{Spec}(Z)$ in favorable cases. The work significantly broadens the scope of boundedness and moduli understanding for Fano surfaces in positive and imperfect fields, with implications for the study of generic fibres of Mori fibre spaces in higher dimensions.

Abstract

We prove several boundedness statements for geometrically integral normal del Pezzo surfaces $X$ over arbitrary fields. We give an explicit sharp bound on the irregularity if $X$ is canonical or regular. In particular, we show that wild canonical del Pezzo surfaces exist only in characteristic 2. As an application, we deduce that canonical del Pezzo surfaces form a bounded family over $\mathbb{Z}$, generalising work of Tanaka. More generally, we prove the BAB conjecture on the boundedness of $\varepsilon$-klt del Pezzo surfaces over arbitrary fields of characteristic different from 2, 3, and 5.

Bounding geometrically integral del Pezzo surfaces

TL;DR

This paper advances the understanding of boundedness for del Pezzo surfaces over arbitrary fields by proving sharp irregularity bounds for geometrically integral normal locally complete intersection del Pezzo surfaces and showing wild canonical del Pezzo surfaces occur only in characteristic . It extends Tanaka’s boundedness results to the canonical case, and establishes BAB-type boundedness for geometrically integral canonical and certain -klt del Pezzo surfaces in characteristic not equal to , using a combination of Frobenius techniques, -torsors, and effective vanishing. Key contributions include effective vanishing and very ampleness results for anti-pluricanonical systems, a detailed analysis of irregularity via the Frobenius length and δ-invariants, and explicit volume and index bounds that lead to bounded families over in favorable cases. The work significantly broadens the scope of boundedness and moduli understanding for Fano surfaces in positive and imperfect fields, with implications for the study of generic fibres of Mori fibre spaces in higher dimensions.

Abstract

We prove several boundedness statements for geometrically integral normal del Pezzo surfaces over arbitrary fields. We give an explicit sharp bound on the irregularity if is canonical or regular. In particular, we show that wild canonical del Pezzo surfaces exist only in characteristic 2. As an application, we deduce that canonical del Pezzo surfaces form a bounded family over , generalising work of Tanaka. More generally, we prove the BAB conjecture on the boundedness of -klt del Pezzo surfaces over arbitrary fields of characteristic different from 2, 3, and 5.
Paper Structure (17 sections, 40 theorems, 58 equations)

This paper contains 17 sections, 40 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Geometric reducedness and normality
  5. Del Pezzo surfaces
  6. Bounds on the anticanonical volume and effective very ampleness
  7. Bounding volumes
  8. Effective Kodaira vanishing and very ampleness on del Pezzo surfaces
  9. Bounds on the irregularity
  10. A bound on $\gamma(X/k)$ for geometrically integral varieties
  11. Anti-pluricanonical maps of non-normal del Pezzo surfaces
  12. Irregularity of geometrically integral l.c.i. del Pezzo surfaces
  13. Refinements in the regular and canonical case
  14. On the BAB conjecture for surfaces over arbitrary fields
  15. Boundedness of geometrically integral canonical del Pezzo surfaces
  16. ...and 2 more sections

Key Result

theorem 1

Let $X$ be a geometrically integral normal locally complete intersection del Pezzo surface over a field $k$ of characteristic $p$. If $h^1(X, \mathcal{O}_X) \neq 0$, then $k$ is an imperfect field, $\rho(X)=1$, and either

Theorems & Definitions (85)

  • theorem 1
  • theorem 2
  • definition 1
  • theorem 3: PW
  • lemma 1: BT22
  • proposition 1
  • proof
  • definition 2
  • proposition 2
  • proof
  • ...and 75 more