The Dolgachev surface

Abstract

We prove that the Dolgachev surface $E(1{)}_{2,3}$ admits a handlebody decomposition without 1- and 3-handles, and we draw the explicit picture of this handlebody. We also locate a “cork” inside of $E(1{)}_{2,3}$, so that $E(1{)}_{2,3}$ is obtained from $E(1)$ by twisting along this cork.