On a Variant of Bertini’s Theorem and Generators of Ideals of a Polynomial Ring with Monic Polynomials

In this paper we discuss different versions of Bertini’s theorem and prove a variant of a theorem of Bertini. We also prove the following theorem. Suppose A is a semilocal regular reduced affine algebra over an algebraically closed field of characteristic zero such that every maximal ideal in A[T ] is complete intersection, with dim(A) = n > 0. Suppose I is an ideal of A[T ], with height(I) = n+ 1. Assume co-normal module I/I is generated by n+ 1 elements over A[T ]/I. Then we can find a minimal set of generators of I with a monic polynomial.