Some (big) irreducible components of the moduli space of minimal surfaces of general type with p_g=q=1 and K^2=4

This paper is devoted to the irregular surfaces of general type with the smallest possible genus and irregularity. We consider the still unexplored case of the minimal surfaces whose canonical system has self-intersection 4, classifying those whose Albanese morphism has general fibre of genus 2, and such that the direct image of the bicanonical sheaf splits as sum of line bundles. We find 8 unirational families, and we prove that they give irreducible components of the moduli space of surfaces of general type. This is unexpected, because the assumption on the direct image of the bicanonical sheaf is a priori only a closed condition. One more unexpected property is that all these components have dimension bigger than the expected one.