Effective Bertini theorem and formulas for multiplicity and the local Łojasiewicz exponent

Abstract

The classical Bertini theorem on generic intersection of an algebraic set with hyperplanes states the following: Let X be a nonsingular closed subvariety ofPkn, where k is an algebraically closed field. Then there exists a hyperplaneH⊂Pknnot containing X and such that the schemeH∩Xis regular at every point. Furthermore, the set of hyperplanes with this property forms an open dense subset of the complete linear system|H|considered as a projective space. We show that one can effectively indicate a finite family of hyperplanes H such that at least one of them satisfies the assertion of the Bertini theorem, provided the characteristic of the field k is equal to zero. As an application of the method used in the proof we give effective formulas for the multiplicity and the Łojasiewicz exponent of polynomial mappings.