Unprojection and deformations of tertiary Burniat surfaces

TL;DR

This paper constructs a new 4-dimensional family of surfaces of general type with specific invariants and fundamental group, extending Burniat surfaces, and describes their universal covers as complete intersections with group actions.

Contribution

It introduces a novel family of surfaces with particular invariants and fundamental group, constructed via unprojection and covering techniques, including their universal covers.

Findings

Constructed a 4-dimensional family of surfaces with p_g=0, K^2=3, and fundamental group Z/2xQ_8.

Embedded these surfaces in Fano 3-folds using parallel unprojection.

Described universal covers as complete intersections in ( ext{P}^1)^4 with group actions.

Abstract

We construct a 4-dimensional family of surfaces of general type with p_g=0 and K^2=3 and fundamental group Z/2xQ_8, where Q_8 is the quaternion group. The family constructed contains the Burniat surfaces with K^2=3. Additionally, we construct the universal coverings of the surfaces in our family as complete intersections on (\PP^1)^4 and we also give an action of Z/2xQ_8 on (\PP^1)^4 lifting the natural action on the surfaces. The strategy is the following. We consider an \'etale (Z/2)^3-cover T of a surface with p_g=0 and K^2=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 universal covers above.