Surfaces of general type with 𝑝_{𝑔}=𝑞=1,𝐾²=8 and bicanonical map of degree 2

Abstract

We classify the minimal algebraic surfaces of general type with p g = q = 1 , K 2 = 8 p_g=q=1, \; K^2=8 and bicanonical map of degree 2 2 . It will turn out that they are isogenous to a product of curves, i.e. if S S is such a surface, then there exist two smooth curves C , F C, \; F and a finite group G G acting freely on C × F C \times F such that S = ( C × F ) / G S = (C \times F)/G . We describe the C , F C, \; F and G G that occur. In particular the curve C C is a hyperelliptic-bielliptic curve of genus 3 3 , and the bicanonical map ϕ \phi of S S is composed with the involution σ \sigma induced on S S by τ × i d : C × F ⟶ C × F \tau \times id: C \times F \longrightarrow C \times F , where τ \tau is the hyperelliptic involution of C C . In this way we obtain three families of surfaces with p g = q = 1 , K 2 = 8 p_g=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 generically smooth, irreducible components of the moduli space M \mathcal {M} of surfaces with p g = q = 1 , K 2 = 8 p_g=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.