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On the coregularity of del Pezzo surfaces with du Val singularities

Konstantin Loginov, Andrey Trepalin

Abstract

We compute the coregularity of del Pezzo surfaces with du Val singularities. To this aim, we study the relation between del Pezzo surfaces of degree $1$ and elliptic fibrations. It turns out that del Pezzo surfaces with positive coregularity correspond to isotrivial elliptic fibrations with some special properties. We also prove results about coregularity of del Pezzo surfaces over non-algebraically closed fields of characteristic $0$. Our results confirm the expectation that "most" del Pezzo surfaces have coregularity $0$, while del Pezzo surfaces with positive coregularity enjoy some special properties.

On the coregularity of del Pezzo surfaces with du Val singularities

Abstract

We compute the coregularity of del Pezzo surfaces with du Val singularities. To this aim, we study the relation between del Pezzo surfaces of degree and elliptic fibrations. It turns out that del Pezzo surfaces with positive coregularity correspond to isotrivial elliptic fibrations with some special properties. We also prove results about coregularity of del Pezzo surfaces over non-algebraically closed fields of characteristic . Our results confirm the expectation that "most" del Pezzo surfaces have coregularity , while del Pezzo surfaces with positive coregularity enjoy some special properties.
Paper Structure (14 sections, 29 theorems, 50 equations)

This paper contains 14 sections, 29 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Contractions
  4. Pairs and singularities
  5. Complements and Calabi--Yau pairs
  6. Dual complex and coregularity
  7. Coregularity
  8. Del Pezzo surfaces
  9. Del Pezzo surfaces over $\mathbb{C}$
  10. Isotrivial fibrations with $j=0$
  11. Isotrivial fibrations with $j=1728$
  12. Del Pezzo surfaces over non-closed fields
  13. Cubic surfaces
  14. Questions

Key Result

Proposition 1.1

Let $\mathbb{K}$ be an algebraically closed field of characteristic $0$. Let $X$ be a smooth del Pezzo surface of degree $d=(-K_X)^2$ defined over $\mathbb{K}$. If $d\geqslant 2$ then $X$ has coregularity $0$. A general smooth del Pezzo surface with $d=1$ has coregularity $0$. For a special smooth d

Theorems & Definitions (75)

  • Proposition 1.1: ALP24
  • Proposition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 65 more