\'Etale cohomology, Lefschetz Theorems and Number of Points of Singular Varieties over Finite Fields

We prove a general inequality for estimating the number of points of arbitrary complete intersections over a finite field. This extends a result of Deligne for nonsingular complete intersections. For normal complete intersections, this inequality generalizes also the classical Lang-Weil inequality. Moreover, we prove the Lang-Weil inequality for affine as well as projective varieties with an explicit description and a bound for the constant appearing therein. We also prove a conjecture of Lang and Weil concerning the Picard varieties and \'etale cohomology spaces of projective varieties. The general inequality for complete intersections may be viewed as a more precise version of the estimates given by Hooley and Katz. The proof is primarily based on a suitable generalization of the Weak Lefschetz Theorem to singular varieties together with some Bertini-type arguments and the Grothendieck-Lefschetz Trace Formula. We also describe some auxiliary results concerning the \'etale cohomology spaces and Betti numbers of projective varieties over finite fields and a conjecture along with some partial results concerning the number of points of projective algebraic sets over finite fields.

[1]  A. Weil.  Numbers of solutions of equations in finite fields , 1949 .

[2]  Jean-Pierre Serre,et al.  Groupes algébriques et corps de classes , 1975 .

[3]  J. Jouanolou,et al.  Théorèmes de Bertini et applications , 1983 .

[4]  C. Hooley,et al.  On the number of points on a complete intersection over a finite field , 1991 .

[5]  G. L..  Collected Papers , 1912, Nature.

[6]  P. Deligne.  La conjecture de Weil. I , 1974 .

[7]  A. Weil.  Sur les critères d'équivalence en géométrie algébrique , 1954 .

[8]  A. Grothendieck,et al.  Cohomologie l-adique et fonctions L , 1977 .

[9]  Jean-Pierre Serre.  Morphismes universels et variété d'Albanese , 1959 .

[10]  Nicholas M. Katz,et al.  Estimates for “singular” exponential sums , 1999 .

[11]  Steven Sperber,et al.  On the degree of the $L$-function associated with an exponential sum , 1988 .

[12]  Marc Perret,et al.  NUMBER OF POINTS OF PRYM VARIETIES OVER FINITE FIELDS , 2006, Glasgow Mathematical Journal.

[13]  A. Weil.  Sur les courbes algébriques et les variétés qui s'en déduisent , 1948 .

[14]  F. Hirzebruch.  Der Satz von Riemann-Roch in Faisceau-theoretischer Formulierung: einige Anwendungen und offene Fragen , 1956 .

[15]  P. Deligne,et al.  Groupes de monodromie en geometrie algebrique , 1972 .

[16]  S. Lang,et al.  NUMBER OF POINTS OF VARIETIES IN FINITE FIELDS. , 1954 .

[17]  Alexander Grothendieck,et al.  Technique de descente et théorèmes d'existence en géométrie algébrique. VI. Les schémas de Picard : propriétés générales , 1962 .

[18]  Nicholas M. Katz,et al.  Sums of Betti Numbers in Arbitrary Characteristic , 2001 .

[19]  Wolfgang M. Schmidt,et al.  A lower bound for the number of solutions of equations over finite fields , 1974 .

[20]  Sudhir R. Ghorpade,et al.  Number of Solutions of Equations over Finite Fields and a Conjecture of Lang and Weil , 2002 .

[21]  D. Northcott,et al.  Lessons on rings, modules and multiplicities , 1968 .

[22]  A. Grothendieck.  Étude locale des schémas et des morphismes de schémas , 1964 .

[23]  G. Lachaud.  Number of points of plane sections and linear codes defined on algebraic varieties , 1996 .

[24]  E. Bombieri,et al.  On the estimation of certain exponential sums , 1995 .

[25]  S. Lang,et al.  Finiteness Theorems in Geometric Classfield Theory , 2000 .

[26]  R. Hartshorne.  Equivalence relations of algebraic cycles and subvarieties of small codimension , 1974 .

[27]  S. Lipman,et al.  Introduction to resolution of singularities , 1974 .

[28]  P. Deligne,et al.  La Conjecture de Weil. II , 1980 .

[29]  Ricardo Garcia Lopez.  On exponential sums , 1997 .

[30]  C. S. Seshardi.  Variété de Picard d'une variété complète , 1962 .

[31]  Alexei N. Skorobogatov,et al.  Exponential sums and rational points on complete intersections , 1990 .

[32]  A. Grothendieck,et al.  Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux, (SAG 2) : séminaire de géométrie algébrique, du Bois Marie, 1962 , 2005 .

[33]  J. Jouanolou.  Cohomologie de quelques schemas classiques et theorie cohomologique des classes de Chern , 1977 .

[34]  Alexander Grothendieck,et al.  Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2) , 1962 .

[35]  S. Abhyankar.  Resolution Of Singularities Of Embedded Algebraic Surfaces , 1966 .

[36]  Serge Lang,et al.  Abelian varieties , 1983 .

[37]  A. Skorobogatov.  Exponential sums, the geometry of hyperplane sections, and some diophantine problems , 1992 .

[38]  Yves Aubry,et al.  A Weil theorem for singular curves , 1996 .

[39]  P. Deligne.  Résumé des premiers exposés de A. Grothendieck , 1972 .

[40]  T. Willmore.  Algebraic Geometry , 1973, Nature.