Polar varieties, Bertini's theorems and number of 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 s  , and let π:V⇢Ps+1(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 Ps+1(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 FqFq-rational point. Finally, for s=r−2s=r−2 and s=r−3s=r−3 we estimate the number of FqFq-rational points and smooth FqFq-rational points of V.