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.