The Joint Reduction Number and Upper Bounds of Hilbert Series of Fiber Cones

Abstract

Let (R,𝔪) be a Cohen-Macaulay local ring of dimension d > 0, I an 𝔪-primary ideal of R and K an ideal containing I. Let a1,…,ad be a joint reduction of (I[d-1]|K), and set L=(a1,…,ad), J=(a1,…,ad-1). When depth G(I) ≥ d-1 and depth FK(I) ≥ d-2, we show that the lengths [Formula: see text], [Formula: see text] and the joint reduction number rL(I|K) are independent of L. In the general case, we give an upper bound of the Hilbert series of FK(I). When depth G(I) ≥ d-1, we also provide a characterization, in terms of the Hilbert series of FK(I), of the condition depth FK(I) ≥ d-1.