Explicit estimates for the number of rational points of singular complete intersections over a finite field

GUILLERMO MATERA; MARIANA PÉREZ; MELINA PRIVITELLI

JOURNAL OF NUMBER THEORY

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2016 vol. 158 p. 54 - 72

0022-314X

Let VcP^n(Fq(Fq) be a complete intersection defined over a finite field Fq of dimension r and singular locus of dimension at most 0leq s leq r-2. We obtain an explicit version of the Hooley-Katz estimate ||V(Fq)|-pr|=O(q(r+s+1)/2), where |V(Fq)| denotes the number of Fq-rational points of V and pr:=|Pr(Fq)|. Our estimate improves all the previous estimates in several important cases. Our approach relies on tools of classical algebraic geometry. A crucial ingredient is a new effective version of the Bertini smoothness theorem, namely an explicit upper bound of the degree of a proper Zariski closed subset of Vc(P^n)^(s+1)(Fq) which contains all the singular linear sections of V of codimension s+1.