Polar varieties, Bertini's theorems and number of points of singular complete intersections over a finite field

Abstract

Let V ⊂ double-struck Pn(double-struck F¯q) be a complete intersection defined over a finite field double-struck Fq of dimension r and singular locus of dimension at most s, and let π :V → double-struck Ps+1 (double-struck F¯q) be a generic linear mapping. We obtain an effective version of the Bertini smoothness theorem concerning π, namely an explicit upper bound of the degree of a proper Zariski closed subset of double-struck Ps+1(double-struck F¯q) which contains all the points defining singular fibers of π. For this purpose we make use of the concept of polar variety associated with the set of exceptional points of π. As a consequence, we obtain results of existence of smooth rational points of V, that is, conditions on q which imply that V has a smooth double-struck Fq-rational point. Finally, for s = r - 2 and s = r - 3 we estimate the number of double-struck Fq-rational points and smooth double-struck Fq-rational points of V.