The moduli space of even surfaces of general type with K2=8, pg=4 and q=0

Even surfaces of general type with K-2 = 8, p(g) = 4 and q = 0 were found by Oliverio [35] as complete intersections of bidegree (6, 6) in a weighted projective space P(1, 1, 2, 3, 3). In this article we prove that the moduli space of even surfaces of general type with K-2 = 8, p(g) = 4 and q =0 consists of two 35-dimensional irreducible components intersecting in a codimension one subset (the first of these components is the closure of the open set considered by Oliverio). All the surfaces in the second component have a singular canonical model, hence we get a new example of a generically nonreduced moduli space. Our result gives a posteriori a complete description of the half-canonical rings of the above even surfaces. The method of proof is, we believe, the most interesting part of the paper. After describing the graded ring of a cone we are able, combining the explicit description of some subsets of the moduli space, some deformation theoretic arguments, and finally some local algebra arguments, to describe the whole moduli space. This is the first time that the classification of a class of surfaces is done using moduli theory: up to now first the surfaces were classified, on the basis of some numerical inequalities, or other arguments, and later on the moduli spaces were investigated. (C) 2013 Elsevier Masson SAS. All rights reserved.