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

Let V⊂Pn(F‾q) be a complete intersection defined over a finite field FqFq of dimension r   and singular locus of dimension at most 0≤s≤r−20≤s≤r−2. We obtain an explicit version of the Hooley–Katz estimate ||V(Fq)|−pr|=O(q(r+s+1)/2)||V(Fq)|−pr|=O(q(r+s+1)/2), where |V(Fq)||V(Fq)| denotes the number of FqFq-rational points of V   and pr:=|Pr(Fq)|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 (Pn)s+1(F‾q) which contains all the singular linear sections of V   of codimension s+1s+1.