Unprojection and deformations of tertiary Burniat surfaces

We construct a 4-dimensional family of surfaces of general type of genus zero, canonical degree 3 and fundamental group G isomorphic to the direct product of the cyclic group of order 2 with the quaternions. The family constructed contains the Burniat surfaces with the same canonical degree. Additionally, we construct the universal coverings of the surfaces in our family as complete intersections on the product of 4 projective lines and we also give an action of G on it lifting the natural action on the surfaces. The strategy is the following. We consider an étale tridouble cover T of a surface of genus zero and canonical degree 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.