Map of Degree 2

We classify the minimal algebraic surfaces of general type with pg = q = 1, K 2 = 8 and bicanonical map of degree 2. It will turn out that they are isogenous to a product of curves, so that 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 σ 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, K 2 = 8 which yield the first known examples of surfaces with these invariants. We compute their dimension, and we show that they are three smooth and irreducible components of the moduli space M of surfaces with pg = q = 1, K 2 = 8. For each of these families, an alternative description as a double cover of the plane is also given, and the index of the paracanonical system is computed.