The classification of double planes of general type with $K^2=8$ and $p_g=0$

Abstract

We study minimal {\em double planes} of general type with $K^2=8$ and $p_g=0$, namely pairs $(S,\sigma)$, where $S$ is a minimal complex algebraic surface of general type with $K^2=8$ and $p_g=0$ and $\sigma$ is an automorphism of $S$ of order 2 such that the quotient $S/\sigma$ is a rational surface. We prove that $S$ is a free quotient $(F\times C)/G$, where $C$ is a curve, $F$ is an hyperelliptic curve, $G$ is a finite group that acts faithfully on $F$ and $C$, and $\sigma$ is induced by the automorphism $\tau\times Id$ of $F\times C$, $\tau$ being the hyperelliptic involution of $F$. We describe all the $F$, $C$ and $G$ that occur: in this way we obtain 5 families of surfaces with $p_g=0$ and $K^2=8$, of which we believe only one was previously known. Using our classification we are able to give an alternative description of these surfaces as double covers of the plane, thus recovering a construction proposed by Du Val. In addition we study the geometry of the subset of the moduli space of surfaces of general type with $p_g=0$ and $K^2=8$ that admit a double plane structure.