Note on Applications of Toric Systems on 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.

arXiv comment24 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

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