Mutations of Numerically Exceptional Collections on Surfaces

TL;DR

This paper investigates the behavior of numerically exceptional collections on rational surfaces, proving transitivity of the braid group action and conditions for collections to be full, with implications for derived categories of these surfaces.

Contribution

It demonstrates the transitivity of the braid group action on maximal numerically exceptional collections on rational surfaces and establishes criteria for these collections to be full.

Findings

Braid group acts transitively on maximal numerically exceptional collections up to isometries and line bundle twists.

Maximal numerically exceptional collections of line bundles are full on certain rational surfaces.

Existence of non-full exceptional collections of maximal length on the blow-up of the plane in 10 points.

Abstract

A conjecture of Bondal-Polishchuk states that, in particular for the bounded derived category of coherent sheaves on a smooth projective variety, the action of the braid group on full exceptional collections is transitive up to shifts. We show that the braid group acts transitively on the set of maximal numerically exceptional collections on rational surfaces up to isometries of the Picard lattice and twists with line bundles. Considering the blow-up of the projective plane in up to 9 points in very general position, these results lift to the derived category. More precisely, we prove that, under these assumptions, a maximal numerically exceptional collection consisting of line bundles is a full exceptional collection and any two of them are related by a sequence of mutations and shifts. The former extends a result of Elagin-Lunts and the latter a result of Kuleshov-Orlov, both concerning del Pezzo surfaces. In contrast, we show in concomitant work [Krah, A Phantom on a Rational Surface, (Preprint)] that the blow-up of the projective plane in 10 points in general position admits a non-full exceptional collection of maximal length consisting of line bundles.