Effective Bertini theorem

Abstract

The classical Bertini theorem on generic intersection of an algebraic set with hyperplanes says that: Let X be a non-singular closed subvariety of $P_k^n$, where $k$ is an algebraically closed field. Then there exists a hyperplane $H\subset P_k^n$, not containing X, and such that the scheme $H\cap X$ is regular at every point. Furthermore, the set of hyperplanes with this property forms an open dense subset of the complete linear system $\left|H\right|$, considered as a projective space. We will show that one can effectively indicate a finite family of hyperplanes H, at least one of which satisfies the assertion of the Bertini theorem, provided that the dimension and degree of the set X are given.