Notes on hyperelliptic fibrations of genus 3, I

We shall study the structure of hyperelliptic fibrations of genus 3, from the view point given by Catanese and Pignatelli in arXiv:math/0503294. In this part I, we shall give a structure theorem for such fibrations for the case of f : S \to B with all fibers 2-connected. We shall also give, for the case of B projective line, sufficient conditions for the existence of such fibrations from the view point of our structure theorem, prove the uniqueness of the deformation type and the simply connectedness of S for some cases, and give some examples including those with minimal regular S with geometric genus 4 and the first Chern number 8. The last example turns out to be a member of the family M_0 given in Bauer--Pignatelli arXiv:math/0603094.