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

ANTONIO CAFURE; GUILLERMO MATERA; MELINA PRIVITELLI

FINITE FIELDS AND THEIR APPLICATIONS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2015 vol. 31 p. 42 - 83

1071-5797

Let V subset P^n(Fq) be a complete intersection defined overa finite field Fq of dimension r and singular locus of dimension at most s, and let pi : V o P^(s+1)(Fq) be a generic linear mapping. We obtain an effective version of the Bertini smoothness theorem concerning pi, namely an explicit upper bound of the degree of a proper Zariski closed subset of P(s+1)(Fq) which contains all the points defining singular fibers of pi. For this purpose we make use of the concept of polar ariety associated with the set of exceptional points of pi. 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 Fq-rational point. Finally, for s = r-2 and s= r-3 we estimate the number of Fq-rational points and smooth Fq-rational points of V .