Some (big) Irreducible Components of the Moduli Space of Minimal Surfaces of General Type

Minimal surfaces of general type with pg = q (i.e with χ(O) = 1, the minimal possible value) have attracted the interest of many authors, but we are very far from a complete classification of them. Gieseker theorem ensures that there are only a finite number of families, but recent results show that the number of this families is huge, at least for the case pg = q = 0 (cf. [PK] for many examples with K 2 = 9). The irregular case is possibly more affordable. There is a complete classification of the case pg = q ≥ 3 ([HP], [Pir], see also [BCP] for more details on what is known on surfaces with χ(O) = 1). We are interested in the case pg = q = 1. A classification of the minimal surfaces of general type with pg = q = 1 and K 2 ≤ 3 has been obtained ([Cat1], [CC1], [CC2], [CP]) by looking at the Albanese morphism, that fibre any surface with q = 1 onto an elliptic curve. In this paper we start the analysis of the next case K = 4, by studying the surfaces whose general Albanese fibre has the minimal