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Dolgachev surfaces are simply connected minimal elliptic surfaces with$p_g=q=0$ and of Kodaira dimension 1. These surfaces were constructed bylogarithmic transformations of rational elliptic surfaces. In this paper, weexplain the construction of Dolgachev surfaces via $\mathbb Q$-Gorensteinsmoothing of singular rational surfaces with two cyclic quotient singularities.This construction is based on the paper by Lee-Park. Also, some exceptionalbundles on Dolgachev surfaces associated with $\mathbb Q$-Gorenstein smoothingare constructed based on the idea of Hacking. In the case if Dolgachev surfaceswere of type $(2,3)$, we describe the Picard group and present an exceptionalcollection of maximal length. Finally, we prove that the presented exceptionalcollection is not full, hence there exist a nontrivial phantom category in thederived category.

Exceptional collections on Dolgachev surfaces associated with degenerations