Exceptional collections on Dolgachev surfaces associated with degenerations

Abstract

Dolgachev surfaces are simply connected minimal elliptic surfaces with $p_g=q=0$ and of Kodaira dimension 1. These surfaces were constructed by logarithmic transformations of rational elliptic surfaces. In this paper, we explain the construction of Dolgachev surfaces via $\mathbb Q$-Gorenstein smoothing of singular rational surfaces with two cyclic quotient singularities. This construction is based on the paper by Lee-Park. Also, some exceptional bundles on Dolgachev surfaces associated with $\mathbb Q$-Gorenstein smoothing are constructed based on the idea of Hacking. 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 exist a nontrivial phantom category in the derived category.