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We settle the first step for the classification of surfaces of general typewith K^2 = 8, p_g = 4 and q = 0, classifying the even surfaces (K is2-divisible).  The first even surfaces of general type with $K^2=8$, $p_g=4$ and $q=0$ werefound by Oliverio as complete intersections of bidegree (6,6) in a weightedprojective space P(1,1,2,3,3).  In this article we prove that the moduli space of even surfaces of generaltype with K^2 = 8, p_g = 4 and q = 0 consists of two 35 -dimensionalirreducible components intersecting in a codimension one subset (the first ofthese components is the closure of the open set considered by Oliverio). Forthe surfaces in the second component the canonical models are always singular,hence we get a new example of generically nonreduced moduli spaces.  Our result gives a posteriori a complete description of the half-canonicalrings of the above even surfaces. The method of proof is, we believe, the mostinteresting part of the paper. After describing the graded ring of a cone weare able, combining the explicit description of some subsets of the modulispace, some deformation theoretic arguments, and finally some local algebraarguments, to describe the whole moduli space. This is the first time that theclassification of a class of surfaces can only be done using moduli theory: upto now first the surfaces were classified, on the basis of some numericalinequalities, or other arguments, and later on the moduli spaces wereinvestigated.

The moduli space of even surfaces of general type with K^2 = 8, p_g = 4 and q = 0