Fibrations of low genus, I

In the present paper we consider fibrations f : S -> B of an algebraic surface over a curve B, with general fibre a curve of genus g. Our main results are: (1) A structure theorem for such fibrations in the case where g = 2. (2) A structure theorem for such fibrations in the case where g = 3, the general fibre is nonhyperelliptic, and each fibre is 2-connected. (3) A theorem giving a complete description of the moduli space of minimal surfaces of general type with p(g) = q = 1, K-S(2) = 3, showing in particular that it has four unirational connected components. (4) Other applications of the two structure theorems. (c) 2006 Elsevier Masson SAS.