Explicit Estimates for the Number of Rational Points of Singular Complete Intersections over a Finite Field

Abstract

Let $V\subset {\mathbb{P}}^n({\overline{F}}_q)$ be a complete intersection defined over a finite field $F_q$ of dimension $r$ and singular locus of dimension at most $0\le s\le r-2$. We obtain an explicit version of the Hooley--Katz estimate $||V(F_q)|-p_r|=\mathcal{O}(q^{(r+s+1)/2})$, where $|V(F_q)|$ denotes the number of $F_q$-rational points of $V$ and $p_r:=|{\mathbb{P}}^r(F_q)|$. 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 $({\mathbb{P}}^n{)}^{s+1}({\overline{F}}_q)$ which contains all the singular linear sections of $V$ of codimension $s+1$.

arXiv admin note: text overlap with arXiv:1209.4938