Abstract

We construct a 4-dimensional family of surfaces of general type with pg=0 and K2=3 and fundamental z/2xqg is the quaternion group. The family constructed contains the Burniat surfaces with K2=3. Additionally, we construct the universal coverings of the surfaces in our family as complete intersections on 4 and we also give an action of Z/2xQ8 on 4 lifting the natural action on the surfaces. The strategy is the following. We consider an étale 3-cover T of a surface with pg=0 and K2=3 and assume that it may be embedded in a Fano 3-fold V. We construct V by using the theory of parallel unprojection. Since V is an Enriques-Fano 3-fold, considering its Fano cover yields the simple description of the above universal covers