Effective Bertini theorems and rational points of complete intersections over a finite field

GUILLERMO MATERA

Córdoba

Congreso; IV Congreso Latinoamericano de Matemáticos; 2012

Universidad Nacional de Córdoba

The classical Bertini theorems assert that, if a given algebraic variety V has a certain property, such as irreducibility or nonsingularity, then a generic positive-dimensional linear section of V has the property too. An effective Bertini theorem provides an upper bound on the degree of the genericity condition underlying the corresponding Bertini theorem. Bertini theorems and effective Bertini theorems play a fundamental role in establishing estimates on the number of rational points of varieties over a finite field (see, e.g., [GhLa02], [CaMa06], [CaMa07]). In this talk we shall discuss effective Bertini theorems for varieties over a finite field, and present a new effective Bertini theorem concerning the singular locus of projective complete intersections. We shall also comment on results of existence and estimates on the number of rational points of projective singular complete intersections over a finite field which are obtained applying these effective Bertini theorems. References. [GhLa02] S. Ghorpade and G. Lachaud, Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields, Mosc. Math.J. 2(3) (2002), 589-631. [CaMa06] A. Cafure and G. Matera, Improved explicit estimates on the number of solutions of equations over a finite field, Finite Fields Appl. 12(2) (2006), 155-185. [CaMa07] A. Cafure and G. Matera, An effective {Bertini} theorem and the number of rational points of a normal complete intersection over a finite field, Acta Arith. 130(1) (2007), 19-35.