Exceptional collections on Dolgachev surfaces associated with degenerations

Abstract

Dolgachev surfaces are simply connected minimal elliptic surfaces with pg=q=0 and of Kodaira dimension 1. These surfaces are constructed by logarithmic transformations of rational elliptic surfaces. In this paper, we explain the construction of Dolgachev surfaces via Q-Gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities. This construction is based on the paper [25]. Also, some exceptional bundles on Dolgachev surfaces associated with Q-Gorenstein smoothing have been constructed based on the idea of Hacking [12]. In the case if Dolgachev surfaces were of type (2,3), we describe the Picard group and present an exceptional collection of maximal length. Finally, we prove that the presented exceptional collection is not full, hence there exists a nontrivial phantom category in the derived category.