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Lines on cubic hypersurfaces over finite fields. (English) Zbl 1448.14022
Bogomolov, Fedor (ed.) et al., Geometry over nonclosed fields. Proceedings of the Simons symposium, March 22–28, 2015. Cham: Springer. Simons Symp., 19-51 (2017).
The purpose of this interesting paper is to study the existence of lines defined over a finite field \({\mathbb F}_q\) on cubic hypersurface \(X\) of dimension \(n\) in finite projective space \({\mathbb P}^{n+1}\) over \({\mathbb F}_q\) and properties of variety of lines \(F(X)\) (if it exists) in some special cases. Such hypersurfaces and varieties over fields have become a very intense area of research in several last decades and one main reason for this is their fundamental role in algebraic geometry. The paper contains the answers to four open problems concerning: lines in smooth cubic hypersurfaces; the proof of the Tate conjecture for \(F(X)\), where \(X \subset {\mathbb P}^{4}_{{\mathbb F}_{2^r}}\) is a smooth cubic hypersurface defined over \({\mathbb F}_{2^r}\); the Picard number of varieties of lines \(F(X)\) in smooth cubic threefold \(X\) and the Picard number of varieties of lines \(F(X)\) in Fermat and Klein cubic threefolds. The authors begin by expressing number of points of projective varieties defined over \({\mathbb F}_q\) by Grothendieck’s Lefschetz trace formula by P. Deligne [Publ. Math., Inst. Hautes Étud. Sci. 43, 273–307 (1973; Zbl 0287.14001)], by trace formula by N. Katz [Lect. Notes Math. 340, 401–438 (1973; Zbl 0275.14015)] and by formulas by S. Galkin and E. Shinder [“The Fano variety of lines and rationality problem for a cubic hypersurface”, Preprint, arXiv:1405.5154]. Short section 3 and long section 4 assemble authors results on cubic surfaces, cubic threefolds and zeta functions of the surface of lines. The proof of the Tate conjecture in characteristic 2 is presented by authors. Existence of lines in various types of cubic threefolds is investigated and is presented. The existance of a line defined over the field \(k\) of definition of a three-dimensional smooth complete intersection \(Y\) of two quadrics (over \(k\)) is the nesessary and sufficient condition that \(Y\) is \(k\)-rational by O. Benoist and O. Wittenberg [“Intermediate Jacobians and rationality over arbitrary fields”, Preprint, arXiv:1909.12668]. Section five assembles authors results about cubic fourfolds, zeta function of the fourfold of lines and existence on lines in the fourfolds over different types of finite fields. The sixth section presents results about lines in cubics of dimension 5 and more. To prove the results authors use Chevalley-Warning theorem, results by H. Esnault [Invent. Math. 151, No. 1, 187–191 (2003; Zbl 1092.14010)], the parameter space of all cubic hypersurfaces in \({\mathbb P}^{n + 1}_{{\mathbb F}_{q}}\) and the incidence variety. For details we must refer to the paper under review.
For the entire collection see [Zbl 1366.14003].

14G15 Finite ground fields in algebraic geometry
14J70 Hypersurfaces and algebraic geometry
14F20 Étale and other Grothendieck topologies and (co)homologies
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
GitHub; Magma
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