An upper bound on the number of rational points of arbitrary projective varieties over finite fields

Couvreur, Alain

doi:10.1090/proc/13015

An upper bound on the number of rational points of arbitrary projective varieties over finite fields
HTML articles powered by AMS MathViewer

by Alain Couvreur

Proc. Amer. Math. Soc. 144 (2016), 3671-3685

DOI: https://doi.org/10.1090/proc/13015

Published electronically: February 12, 2016

PDF | Request permission

Abstract:

We give an upper bound on the number of rational points of an arbitrary Zariski closed subset of a projective space over a finite field $\mathbf {F}_q$. This bound depends only on the dimensions and degrees of the irreducible components and holds for very general projective varieties, even reducible and nonequidimensional. As a consequence, we prove a conjecture of Ghorpade and Lachaud on the maximal number of rational points of an equidimensional projective variety.

References

Yves Aubry and Marc Perret, Coverings of singular curves over finite fields, Manuscripta Math. 88 (1995), no. 4, 467–478. MR 1362932, DOI 10.1007/BF02567835
Yves Aubry and Marc Perret, On the characteristic polynomials of the Frobenius endomorphism for projective curves over finite fields, Finite Fields Appl. 10 (2004), no. 3, 412–431. MR 2067606, DOI 10.1016/j.ffa.2003.09.005
Albrecht Beutelspacher, Blocking sets and partial spreads in finite projective spaces, Geom. Dedicata 9 (1980), no. 4, 425–449. MR 596739, DOI 10.1007/BF00181559
M. Boguslavsky, On the number of solutions of polynomial systems, Finite Fields Appl. 3 (1997), no. 4, 287–299. MR 1478830, DOI 10.1006/ffta.1997.0186
M. Datta and S. Ghorpade. Number of solutions of systems of homogeneous polynomials over finite fields, ArXiv:1507.03029, 2015.
M. Datta and S. Ghorpade. On a conjecture of Tsfasman and an inequality of Serre for the number of points on hypersurfaces over finite fields, ArXiv:1503.03049, 2015.
Pierre Deligne, La conjecture de Weil. I, Inst. Hautes Études Sci. Publ. Math. 43 (1974), 273–307 (French). MR 340258, DOI 10.1007/BF02684373
David Eisenbud and Joe Harris, The geometry of schemes, Graduate Texts in Mathematics, vol. 197, Springer-Verlag, New York, 2000. MR 1730819
William Fulton, Intersection theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Springer-Verlag, Berlin, 1998. MR 1644323, DOI 10.1007/978-1-4612-1700-8
Sudhir R. Ghorpade and Gilles Lachaud, Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields, Mosc. Math. J. 2 (2002), no. 3, 589–631. Dedicated to Yuri I. Manin on the occasion of his 65th birthday. MR 1988974, DOI 10.17323/1609-4514-2002-2-3-589-631
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
Masaaki Homma and Seon Jeong Kim, Sziklai’s conjecture on the number of points of a plane curve over a finite field III, Finite Fields Appl. 16 (2010), no. 5, 315–319. MR 2678619, DOI 10.1016/j.ffa.2010.05.001
Masaaki Homma and Seon Jeong Kim, An elementary bound for the number of points of a hypersurface over a finite field, Finite Fields Appl. 20 (2013), 76–83. MR 3015353, DOI 10.1016/j.ffa.2012.11.002
Dieter Jungnickel, Maximal partial spreads and translation nets of small deficiency, J. Algebra 90 (1984), no. 1, 119–132. MR 757085, DOI 10.1016/0021-8693(84)90202-3
Dieter Jungnickel, Maximal partial spreads and transversal-free translation nets, J. Combin. Theory Ser. A 62 (1993), no. 1, 66–92. MR 1198381, DOI 10.1016/0097-3165(93)90072-G
Gilles Lachaud and Robert Rolland, On the number of points of algebraic sets over finite fields, J. Pure Appl. Algebra 219 (2015), no. 11, 5117–5136 (English, with English and French summaries). MR 3351576, DOI 10.1016/j.jpaa.2015.05.008
Klaus Metsch and L. Storme, Partial $t$-spreads in $\textrm {PG}(2t+1,q)$, Des. Codes Cryptogr. 18 (1999), no. 1-3, 199–216. Designs and codes—a memorial tribute to Ed Assmus. MR 1738667, DOI 10.1023/A:1008305824113
Jean-Pierre Serre, Sur le nombre des points rationnels d’une courbe algébrique sur un corps fini, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 9, 397–402 (French, with English summary). MR 703906
Jean-Pierre Serre, Lettre à M. Tsfasman, Astérisque 198-200 (1991), 11, 351–353 (1992) (French, with English summary). Journées Arithmétiques, 1989 (Luminy, 1989). MR 1144337
Anders Bjært Sørensen, On the number of rational points on codimension-$1$ algebraic sets in $\textbf {P}^n(\textbf {F}_q)$, Discrete Math. 135 (1994), no. 1-3, 321–334. MR 1310889, DOI 10.1016/0012-365X(93)E0009-S
Karl-Otto Stöhr and José Felipe Voloch, Weierstrass points and curves over finite fields, Proc. London Math. Soc. (3) 52 (1986), no. 1, 1–19. MR 812443, DOI 10.1112/plms/s3-52.1.1
Peter Sziklai, A bound on the number of points of a plane curve, Finite Fields Appl. 14 (2008), no. 1, 41–43. MR 2381474, DOI 10.1016/j.ffa.2007.09.004
Koen Thas, On the number of points of a hypersurface in finite projective space (after J.-P. Serre), Ars Combin. 94 (2010), 183–190. MR 2599730