Effective Bertini theorems and rational points of a variety over a finite field

GUILLERMO MATERA

Sao Pedro

Conferencia; XVIII Coloquio Latinoamericano de Álgebra; 2009

Coloquio Latinoamericano de Álgebra

A Bertini theorem asserts that certain properties of an algebraic variety, such as irreducibility or nonsingularity, are preserved by generic positive-dimensional linear sections of the variety. An effective Bertini theorem provides an upper bound on the degree of the genericity condition underlying the corresponding Bertini theorem. In this talk we shall discuss effective Bertini theorems for varieties over a finite field. We shall also comment on results of existence and estimates on the number of rational points of a variety over a finite field which are obtained applying these effective Bertini theorems.