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

TL;DR

This paper provides explicit bounds for counting rational points on singular complete intersections over finite fields, improving previous estimates by leveraging classical algebraic geometry and an effective Bertini theorem.

Contribution

It introduces an explicit version of the Hooley--Katz estimate for singular complete intersections, with improved bounds and a new effective Bertini smoothness theorem.

Findings

Derived explicit bounds for rational points on singular complete intersections.

Improved upon previous estimates in key cases.

Developed a new effective Bertini smoothness theorem.

Abstract

Let $V\subset\mathbb{P}^n(\overline{F}_{\hskip-0.7mm q})$ be a complete intersection defined over a finite field $F_{\hskip-0.7mm q}$ 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 $||V(F_{\hskip-0.7mm q})|-p_r|=\mathcal{O}(q^{(r+s+1)/2})$, where $|V(F_{\hskip-0.7mm q})|$ denotes the number of $F_{\hskip-0.7mm q}$-rational points of $V$ and $p_r:=|\mathbb{P}^r(F_{\hskip-0.7mm q})|$. 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 $(\mathbb P^{n})^{s+1}(\overline{F}_{\hskip-0.7mm q})$ which contains all the singular linear sections of $V$ of codimension $s+1$.