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Geometry and arithmetic of geometrically integral regular del Pezzo surfaces

Fabio Bernasconi, Hiromu Tanaka

TL;DR

The paper delivers a complete classification of geometrically integral regular del Pezzo surfaces that are not geometrically normal over imperfect fields of positive characteristic, revealing that such phenomena occur only in characteristics $p\in\{2,3\}$ and organizing the possibilities into primitive and imprimitive cases with detailed normalisations and conic-bundle structures. It then translates these geometric classifications into arithmetic consequences relevant to threefold del Pezzo fibrations, proving that terminal threefolds over algebraically closed fields admit a section and deriving rationality criteria when the base is rational and the fiber has anticanonical degree at least five. The work includes explicit constructions (tables and examples) that realise all listed cases, and develops robust rationality and point-existence results over $C_1$-fields, as well as unirationality results for low-degree del Pezzo surfaces. Overall, the results extend classical insights on del Pezzo surfaces to imperfect, positive-characteristic settings, with significant implications for the arithmetic of 3-fold fibrations in the MMP framework.

Abstract

We classify geometrically integral regular del Pezzo surfaces which are not geometrically normal over imperfect fields of positive characteristic. Based on this classification, we show that a three-dimensional terminal del Pezzo fibration onto a curve over an algebraically closed field always admits a section. Moreover, we prove that the total space is rational if the base curve is rational and the anticanonical degree of a fibre is at least five.

Geometry and arithmetic of geometrically integral regular del Pezzo surfaces

TL;DR

The paper delivers a complete classification of geometrically integral regular del Pezzo surfaces that are not geometrically normal over imperfect fields of positive characteristic, revealing that such phenomena occur only in characteristics and organizing the possibilities into primitive and imprimitive cases with detailed normalisations and conic-bundle structures. It then translates these geometric classifications into arithmetic consequences relevant to threefold del Pezzo fibrations, proving that terminal threefolds over algebraically closed fields admit a section and deriving rationality criteria when the base is rational and the fiber has anticanonical degree at least five. The work includes explicit constructions (tables and examples) that realise all listed cases, and develops robust rationality and point-existence results over -fields, as well as unirationality results for low-degree del Pezzo surfaces. Overall, the results extend classical insights on del Pezzo surfaces to imperfect, positive-characteristic settings, with significant implications for the arithmetic of 3-fold fibrations in the MMP framework.

Abstract

We classify geometrically integral regular del Pezzo surfaces which are not geometrically normal over imperfect fields of positive characteristic. Based on this classification, we show that a three-dimensional terminal del Pezzo fibration onto a curve over an algebraically closed field always admits a section. Moreover, we prove that the total space is rational if the base curve is rational and the anticanonical degree of a fibre is at least five.
Paper Structure (33 sections, 48 theorems, 131 equations)

This paper contains 33 sections, 48 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Regular del Pezzo surfaces
  5. Classification of geometrically integral regular del Pezzo surfaces
  6. Primitive case
  7. Proof of Step 1.
  8. Proof of Step 2.
  9. Proof of Step 3.
  10. Proof of Step 4.
  11. Imprimitive case
  12. Classification: tables and proof
  13. Proof of Step 1.
  14. Proof of Step 2.
  15. Proof of Step 3.
  16. ...and 18 more sections

Key Result

Theorem 1.1

Let $k$ be a field of characteristic $p>0$. Let $X$ be a regular del Pezzo surface over $k$. Assume that $X$ is geometrically integral and not geometrically normal. Then $p \in \{2, 3\}$ and $X$ belongs to one of the families in table p=3 and table p=2 of thm: class_dP.

Theorems & Definitions (112)

  • Theorem 1.1: \ref{['thm: class_dP']}
  • Theorem 1.2: \ref{['p deg7']}, \ref{['t rationality']}, \ref{['them: Enriques']}
  • Theorem 1.3: \ref{['thm: rat_min_MFS']}
  • Theorem 1.4: \ref{['thm: existence_section']}, cf. \ref{['r non geom int']}(4)
  • Theorem 1.5
  • Corollary 1: \ref{['ss rat pt']}
  • Proposition 1
  • proof
  • Lemma 1
  • Lemma 1
  • ...and 102 more