Note on Applications of Toric Systems on Surfaces

TL;DR

This paper explores applications of toric systems on surfaces, linking exceptional collections to surface rationality, and addressing conjectures on exceptional sheaves on weak del Pezzo surfaces.

Contribution

It applies toric system techniques to connect exceptional collections with surface rationality and proposes partial solutions to conjectures on exceptional sheaves.

Findings

Full exceptional collections imply rationality for small Picard rank surfaces.

Equivalences of various notions of cyclic strong exceptional collections are established.

Partial progress on a conjecture regarding exceptional sheaves on weak del Pezzo surfaces.

Abstract

In this note we apply the techniques of the toric systems introduced by Hille-Perling to several problems on smooth projective surfaces: We showed that the existence of full exceptional collection of line bundles implies the rationality for small Picard rank surfaces; we proved equivalences of several notions of cyclic strong exceptional collection of line bundles; we also proposed a partial solution to a conjecture on exceptional sheaves on weak del Pezzo surfaces.