Powers of generic ideals and the weak Lefschetz property for powers of some monomial complet intersections

摘要

Given an ideal I = (f(1) ... , f(r)) in C[x(1), ... , x(n),] generated by forms of degree d, and an integer k 1, how large can the ideal I-k be, i.e., how small can the Hilbert function of C[x(1), ... , x(n)] / I-k be? If r = n the smallest Hilbert function is achieved by any complete intersection, but for r n, the question is in general very hard to answer. We study the problem for r = n + 1, where the result is known for k = 1. We also study a closely related problem, the Weak Lefschetz property, for S/I-k, where I is the ideal generated by the d'th powers of the variables. (C) 2017 Elsevier Inc. All rights reserved.