The moduli space of surfaces with K^2=6 and p_g=4

In this paper we answer a question posed by Horikawa in 1978, who studied (in Invent. Math. 47) the moduli space of the regular minimal surfaces of general type with pg=4 and K^2=6, and showed that it is composed of 11 locally closed strata building up 4 irreducible components and having at most 3 connected components. We prove that the number of connected components is at most two. The main new idea is to analyse the strata in the moduli space where the canonical divisor is 2-divisible on the canonical model (as a Weil divisor). In this way we obtain a semicanonical ring which is a Gorenstein ring of codimension 4 for the surfaces of "type IIIb" and of codimension 1 for "type II". We describe a flat family with central fibre of type IIIb and general fibre of type II by studying the flat deformations of these rings. We use one of the formats introduced by Dicks and Reid for Gorenstein rings of codimension 4, the one of 4x4 Pfaffians of antisymmetric extrasymmetric 6x6 matrices.