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Atomic decompositions for derived categories of G-surfaces

Alexey Elagin, Julia Schneider, Evgeny Shinder

TL;DR

The paper constructs canonical G-equivariant semi-orthogonal decompositions for derived categories of smooth projective surfaces, showing compatibility with minimal model program operations and thereby confirming Kontsevich’s conjecture in dimension two. It introduces G-atoms and a mutation-stable atomic theory, extends it to arithmetic settings via Galois descent, and develops a three-block del Pezzo decomposition framework to control the pieces across Sarkisov links. The approach yields concrete connections between birational geometry and derived categories, providing birational invariants, rationality criteria, and a principled framework for descent to non-closed fields. It also links atomic decompositions to classical arithmetic obstructions (Amitsur, Brauer data) and to geometric notions like G-linearisability, with broad implications for birational classification of geometrically rational surfaces.

Abstract

We construct canonical semi-orthogonal decompositions for derived categories of smooth projective surfaces. These decompositions are compatible with the operations in the minimal model program, such as blow-ups and conic bundles. Therefore our construction confirms a conjecture of Kontsevich in dimension two. We work in the G-equivariant setting and over an arbitrary perfect field, and canonical decompositions are consistent with group change and algebraic field extensions. Our method is based on the G-minimal model program for surfaces and on the Sarkisov link factorisation of birational maps between Mori fibre spaces. We characterise rationality of surfaces, and in certain cases, birationality between surfaces in terms of the pieces of these decompositions, which we call atoms.

Atomic decompositions for derived categories of G-surfaces

TL;DR

The paper constructs canonical G-equivariant semi-orthogonal decompositions for derived categories of smooth projective surfaces, showing compatibility with minimal model program operations and thereby confirming Kontsevich’s conjecture in dimension two. It introduces G-atoms and a mutation-stable atomic theory, extends it to arithmetic settings via Galois descent, and develops a three-block del Pezzo decomposition framework to control the pieces across Sarkisov links. The approach yields concrete connections between birational geometry and derived categories, providing birational invariants, rationality criteria, and a principled framework for descent to non-closed fields. It also links atomic decompositions to classical arithmetic obstructions (Amitsur, Brauer data) and to geometric notions like G-linearisability, with broad implications for birational classification of geometrically rational surfaces.

Abstract

We construct canonical semi-orthogonal decompositions for derived categories of smooth projective surfaces. These decompositions are compatible with the operations in the minimal model program, such as blow-ups and conic bundles. Therefore our construction confirms a conjecture of Kontsevich in dimension two. We work in the G-equivariant setting and over an arbitrary perfect field, and canonical decompositions are consistent with group change and algebraic field extensions. Our method is based on the G-minimal model program for surfaces and on the Sarkisov link factorisation of birational maps between Mori fibre spaces. We characterise rationality of surfaces, and in certain cases, birationality between surfaces in terms of the pieces of these decompositions, which we call atoms.

Paper Structure

This paper contains 28 sections, 72 theorems, 197 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Atomic decompositions for $G$-surfaces
  3. Applications in the arithmetic case
  4. Preliminaries
  5. Group actions and $G$-surfaces
  6. Birational geometry of $G$-surfaces
  7. Derived contractions and MMP contractions
  8. Divisor classes on surfaces
  9. Semi-orthogonal decompositions and their descent
  10. Preliminaries on triangulated categories
  11. Semi-orthogonal decompositions and their mutations
  12. Exceptional objects
  13. Serre functor
  14. Preliminaries on derived categories of coherent sheaves
  15. Three-block semi-orthogonal decompositions for del Pezzo surfaces
  16. ...and 13 more sections

Key Result

Theorem 1.3

For every group $G$ there is an explicit atomic theory for $G-\mathop{\mathrm{\mathcal{V}ar}}\nolimits_{{\Bbbk}, \le 2}^{\mathop{\mathrm{\mathrm{MMP}}}\nolimits}$. Furthermore, the constructed theories are compatible with passing to subgroups $H \subset G$.

Figures (1)

  • Figure 1: Affine quivers $\tilde{D}_4$ and $\tilde{A}_3$

Theorems & Definitions (200)

  • Conjecture 1.1: Kontsevich
  • Definition 1.2: see Definition \ref{['def_GSAT']}
  • Theorem 1.3: see Theorem \ref{['thm:atoms-surfaces']}
  • Example 1.4
  • Proposition 1.5: see Proposition \ref{['prop_Z']}
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8: see Theorem \ref{['thm:atoms-surfaces2']}
  • Proposition 1.9: see Theorem \ref{['thm:rational-atoms']}
  • Proposition 1.10: Proposition \ref{['prop:bir-rich-non-closed']}
  • ...and 190 more