Geometrically, the main goal of this thesis is to refine the classification of minimal surfaces S with K2S = 7 and pg = 4 due to Ingrid Bauer and published in her monograph Surfaces with K2S = 7 and pg = 4 (cf. [Bauer]). She found that they belong to 10 families according to the behaviour of the canonical map φKS . The 10 families form 3 irreducible components of moduli, but the details of how this happens remained unknown except for a few particular cases. Our treatment consists in studying the abstract canonical model Proj R(S, KS), where R(S;KS) := Ɵn≥0 nC0 H0(S;OS(nKS)) is the pluricanonical ring. Except when |KS| is base point free, these rings are Gorenstein of codimension ≥4. We show that the only previously known deformation family of such rings (constructed by Bauer, Catanese and Pignatelli in [Bauer et al]) relating the 2 families with φKS birational can be recovered using basic arguments about halfcanonical curves. Our techniques also allow us to construct new explicit at families for cases on which φKS is not birational. In particular, we construct a 1-parameter at family of Gorenstein rings with general fibre of codimension 4 and special fibre of codimension 6. At the end we discuss possible applications of our methods to the cases on which |KS| defines a 2-to-1 map to a quadratic surface. We conjecture that the moduli space of surfaces with K2S = 7 and pg = 4 is connected.