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Geometry and arithmetic of primary Burniat surfaces. (English) Zbl 1390.14116

A Burniat surface \(S\) is a bidouble cover of a degree 6 del Pezzo surface of a certain type, and is said to be primary if it is smooth. They were constructed by P. Burniat [Ann. Mat. Pura Appl. (4) 71, 1–24 (1966; Zbl 0144.20203)]. A completely different construction, as a quotient of a certain hypersurface \(X\) in the product \(E_1\times E_2\times E_3\) of three elliptic curves, was given in [M. Inoue, Tokyo J. Math. 17, No. 2, 295–319 (1994; Zbl 0836.14020)] it was not until 2011 that the first author and F. Catanese [in: Classification of algebraic varieties. Based on the conference on classification of varieties, Schiermonnikoog, Netherlands, May 2009. Zürich: European Mathematical Society (EMS). 49–76 (2011; Zbl 1264.14052)] showed that Inoue’s construction and Burniat’s really give precisely the same surfaces.
The present paper exploits this coincidence in two ways: the authors describe the moduli space, and for a Burniat surface \(S\) defined over \({\mathbb Q}\) they give a detailed description of the set \(S({\mathbb Q})\) of rational points.
The moduli space of Burniat surfaces was studied in M. Mendes Lopes and R. Pardini [Topology 40, No. 5, 977–991 (2001; Zbl 1072.14522)] but using only Burniat’s approach. With the additional help given by Inoue’s alternative, in this paper the authors exhibit the moduli space completely explicitly, giving equations for it in \({\mathbb A}^6\), and stratify it according to the automorphism group of \(S\). In particular they show that it is an irreducible rational variety of dimension 4: this, and some other results were earlier obtained in a less detailed form by the first author and Catanese [loc. cit].
Being of general type, Burniat surfaces are expected to satisfy Lang’s conjecture: in particular, if \(S\) is defined over \({\mathbb Q}\) then \(S({\mathbb Q})\) should be contained in a proper subvariety. The Zariski closure of \(S({\mathbb Q})\) thus consists of, at most, some curves of genus 0 or 1 and finitely many other points. The second part of the paper shows how to conpute these loci in the slightly special case where everything (the \(E_i\), not just \(S\)) are defined over \({\mathbb Q}\). For the curves of genus 0 or 1, one considers their covers in \(X\), which map to each \(E_i\) and are thus elliptic curves isogenous to at least one of the \(E_i\). There are several cases, depending on how many of these maps are non-constant. For the other points, one needs to find rational points on a Galois twist of \(X\), which depend on the nature of the \(E_i({\mathbb Q})\) so again there are several cases.

MSC:

14J29 Surfaces of general type
14G05 Rational points
14G25 Global ground fields in algebraic geometry
14J10 Families, moduli, classification: algebraic theory
14J50 Automorphisms of surfaces and higher-dimensional varieties
14K12 Subvarieties of abelian varieties
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