×

zbMATH — the first resource for mathematics

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].

MSC:
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)
Software:
GitHub; Magma
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] A. Adler, Some integral representations of \(\(PSL_2(\textbf{{F}}_p)\)\) and their applications. J. Algebra 72, 115-145 (1981) · Zbl 0479.20017
[2] A. Altman, S. Kleiman, Foundations of the theory of Fano schemes. Compos. Math. 34, 3-47 (1977) · Zbl 0414.14024
[3] A. Beauville, Variétés de Prym et jacobiennes intermédiaires. Ann. Sci. Éc. Norm. Sup. 10, 309-391 (1977) · Zbl 0368.14018
[4] A. Beauville, Les singularités du diviseur Theta de la jacobienne intermédiaire de l’hypersurface cubique dans \(\(\textbf{{P}}^4\)\), in \(Algebraic Threefolds, Varenna 1981\), Lecture Notes in Mathematics, vol. 947 (Springer, Heidelberg, 1982), pp. 190-208
[5] E. Bombieri, H.P.F. Swinnerton-Dyer, On the local zeta function of a cubic threefold. Ann. Scuola Norm. Sup. Pisa 21, 1-29 (1967) · Zbl 0153.50501
[6] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symbolic Comput. 24, 235-265 (1997) · Zbl 0898.68039
[7] S. Casalaina-Martin, R. Laza, The moduli space of cubic threefolds via degenerations of the intermediate Jacobian. J. reine angew. Math. 663, 29-65 (2009) · Zbl 1248.14041
[8] F. Charles, A. Pirutka, La conjecture de Tate entière pour les cubiques de dimension quatre. Compos. Math. 151, 253-264 (2015) · Zbl 1316.14084
[9] C.H. Clemens, P.A. Griffiths, The intermediate Jacobian of the cubic threefold. Ann. Math. 95, 281-356 (1972) · Zbl 0214.48302
[10] P. Deligne, La conjecture de Weil I. Publ. Math. Inst. Hautes Études Sci. 43, 273-308 (1974) · Zbl 0287.14001
[11] A.-S. Elsenhans, J. Jahnel, On the characteristic polynomial of the Frobenius on étale cohomology. Duke Math. J. 164, 2161-2184 (2015) · Zbl 1348.14056
[12] H. Esnault, Varieties over a finite field with trivial Chow group of \(\(0\)\)-cycles have a rational point. Invent. Math. 151, 187-191 (2003) · Zbl 1092.14010
[13] N. Fakhruddin, C.S. Rajan, Congruences for rational points on varieties over finite fields. Math. Ann. 333, 797-809 (2005) · Zbl 1083.14022
[14] S. Galkin, E. Shinder, The Fano variety of lines and rationality problem for a cubic hypersurface. eprint arXiv:1405.5154v2 · Zbl 1408.14068
[15] N.E. Hurt, in \(Many Rational Points: Coding Theory and Algebraic Geometry\), Mathematics and its Applications, vol. 564 (Kluwer, Dordrecht, 2003) · Zbl 1072.11042
[16] E.W. Howe, K.E. Lauter, J. Top, Pointless curves of genus three and four, in \(Arithmetic, Geometry and Coding Theory (AGCT 2003)\), Séminaries & Congres, vol. 11 (société mathématique de France, Paris, 2005), pp. 125-141 · Zbl 1116.14010
[17] E.W. Howe, H.J. Zhu, On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field. J. Number Theory 92, 139-163 (2002) · Zbl 0998.11031
[18] N. Katz, Une formule de congruence pour la fonction \(\(ζ \)\), Exposé XXII, in \(Groupes de Monodromie en Géométrie Algébrique (SGA 7 II)\), Lecture Notes in Mathematics, vol. 340 (Springer, Heidelberg, 1973), pp. 401-438
[19] K. Kedlaya, Effective \(\(p\)\)-adic cohomology for cyclic cubic threefolds, in \(Computational Algebraic and Analytic Geometry\), Contemporary Mathematics, vol. 572 (American Mathematical Society, Providence, 2012), pp. 127-171 · Zbl 1317.14049
[20] A. Kouvidakis, G. van der Geer, A note on Fano surfaces of nodal cubic threefolds, in \(Algebraic and Arithmetic Structures of Moduli Spaces, Sapporo 2007\), Advanced Studies in Pure Mathematics, vol. 58 (Mathematical Society of Japan, Tokyo, 2010), pp. 27-45 · Zbl 1215.14045
[21] S. Lang, \(Elliptic Functions\) (Addison-Wesley Publishing Co., Inc., Reading, 1973)
[22] J.S. Milne, in \(Étale Cohomology\), Princeton Mathematical Series, vol. 33 (Princeton University Press, Princeton, 1980)
[23] D. Mumford, in \(Abelian Varieties\), Tata Institute of Fundamental Research Studies in Mathematics, vol. 5 (Oxford University Press, Oxford, 1970) · Zbl 0223.14022
[24] J.P. Murre, Some results on cubic threefolds, in \(Classification of Algebraic Varieties and Compact Complex Manifolds\), Lecture Notes in Mathematics, vol. 412 (Springer, Berlin, 1974), pp. 140-160
[25] X. Roulleau, On the Tate conjecture for the Fano surfaces of cubic threefolds. J. Number Theory 133, 2320-2323 (2013) · Zbl 1285.14021
[26] T. Shioda, T. Katsura, On Fermat varieties. Tôhoku Math. J. 31, 97-115 (1979) · Zbl 0415.14022
[27] J.H. Silverman, in \(Advanced Topics in The Arithmetic of Elliptic Curves\), Graduate Texts in Mathematics, vol. 151 (Springer, New York, 1994) · Zbl 0911.14015
[28] J.H. Silverman, J. Tate, in \(Rational Points on Elliptic Curves\), Undergraduate Texts in Mathematics (Springer, New York, 1992)
[29] J. Tate, Algebraic cycles and poles of zeta functions, in \(Arithmetical Algebraic Geometry, Purdue University 1963\), ed. by O. Schilling (Harper and Row, New York, 1965), pp. 93-110
[30] A. Weil, Numbers of solutions of equations in finite fields. Bull. Amer. Math. Soc. 55, 497-508 (1949) · Zbl 0032.39402
[31] Computer computations, https://github.com/alaface/CubLin · Zbl 0296.68050
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.