Surfaces of general type with pg = q = 1, K2 = 8 and bicanonical map of degree 2

We classify the minimal algebraic surfaces of general type with p g = q = 1, K2 = 8 and bicanonical map of degree 2. It will turn out that they are isogenous to a product of curves, i.e. if S is such a surface, then there exist two smooth curves C, F and a finite group G acting freely on C × F such that S = (C × F)/G. We describe the C, F and G that occur. In particular the curve C is a hyperelliptic-bielliptic curve of genus 3, and the bicanonical map φ of S is composed with the involution a induced on S by τ × id: C × F → C × F, where τ is the hyperelliptic involution of C. In this way we obtain three families of surfaces with pg = q = 1, K2 = 8 which yield the first-known examples of surfaces with these invariants. We compute their dimension and we show that they are three generically smooth, irreducible components of the moduli space M of surfaces with pg = q = 1, K 2 = 8. Moreover, we give an alternative description of these surfaces as double covers of the plane, recovering a construction proposed by Du Val. © 2005 American Mathematical Society.