Note on Applications of Toric Systems on Surfaces
Algebraic Geometry
2017-11-28 v2
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.
Cite
@article{arxiv.1710.05274,
title = {Note on Applications of Toric Systems on Surfaces},
author = {Shizhuo Zhang},
journal= {arXiv preprint arXiv:1710.05274},
year = {2017}
}
Comments
24 pages, preliminary sections 2-5 are overlap with the earlier work arXiv:1710.03972. Section 7.1 added: improved a theorem on characterization of quadrics by full exceptional collection via pseudo anti-canonical height. Comments are welcome