The classification of double planes of general type with K2=8 and pg=0

Abstract

We study minimal double planes of general type with K 2 =8 and p g =0, namely pairs ( S , σ ), where S is a minimal complex algebraic surface of general type with K 2 =8 and p g =0, and σ is an automorphism of S of order 2 such that the quotient S / σ is a rational surface. We prove that S is a free quotient ( F × 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 σ is induced by the automorphism τ ×Id of F × C , τ 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.