Atomic decompositions for derived categories of G-surfaces

TL;DR

This paper develops canonical semi-orthogonal decompositions for derived categories of G-surfaces, confirming a conjecture of Kontsevich in dimension two, and relates these decompositions to surface rationality and birationality.

Contribution

It introduces G-equivariant semi-orthogonal decompositions compatible with the minimal model program for surfaces, confirming Kontsevich's conjecture in dimension two.

Findings

Constructed canonical decompositions compatible with blow-ups and conic bundles.

Characterized rationality and birationality of surfaces via decomposition atoms.

Confirmed Kontsevich's conjecture for G-surfaces in dimension two.

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.