Bicanonical map of surfaces with x = 1 fibered by hyperelliptic curves of genus 3

We investigate surfaces of general type S with x(S) = 1 that admit a fibration ƒ : S → ℙ1 whose general fiber is a hyperelliptic curve of genus 3. We show that the bicanonical map of S factors through a generically finite rational map of degree two onto a ruled surface. Together with a previous result, this yields a characterization of surfaces with x = 1 whose bicanonical map factors through a 2 : 1 map onto a ruled surface in terms of the existence of fibrations of small genus.Peer Reviewe