Geometry and arithmetic of primary Burniat surfaces

TL;DR

This paper investigates the geometry and arithmetic of primary Burniat surfaces, providing explicit descriptions of their moduli space, automorphism groups, genus 1 curves, and methods for analyzing rational points over erential fields.

Contribution

It offers a new explicit description of the moduli space of primary Burniat surfaces and details their automorphism groups and rational points analysis.

Findings

Explicit moduli space description for primary Burniat surfaces

Classification of automorphism groups of these surfaces

Methods for finding rational points on specific surfaces

Abstract

We study the geometry and arithmetic of so-called primary Burniat surfaces, a family of surfaces of general type arising as smooth bidouble covers of a del Pezzo surface of degree 6 and at the same time as \'etale quotients of certain hypersurfaces in a product of three elliptic curves. We give a new explicit description of their moduli space and determine their possible automorphism groups. We also give an explicit description of the set of curves of geometric genus 1 on each primary Burniat surface. We then describe how one can try to obtain a description of the set of rational points on a given primary Burniat surface $S$ defined over $\mathbb Q$. This involves an explicit description of the relevant twists of the \'etale covering of $S$ coming from the second construction mentioned above and methods for finding the set of rational points on a given twist.