Uniform Harbourne–Huneke bounds via flat extensions

Abstract

Over an arbitrary field F, Harbourne [3] conjectured that the symbolic power I(N(r−1)+1)⊆Ir for all r>0 and all homogeneous ideals I in S=F[PN]=F[x0,…,xN]. The conjecture has been disproven for select values of N≥2: first by Dumnicki, Szemberg, and Tutaj-Gasińska in characteristic zero [7], and then by Harbourne and Seceleanu in positive characteristic [13]. However, the ideal containments above do hold when, e.g., I is a monomial ideal in S[3, Ex. 8.4.5].As a sequel to [21], we present criteria for containments of type I(N(r−1)+1)⊆Ir for all r>0 and certain classes of ideals I in a prodigious class of normal rings. Of particular interest is a result for monomial primes in tensor products of affine semigroup rings. Indeed, we explain how to give effective multipliers N in several cases including: the D-th Veronese subring of any polynomial ring F[x1,…,xn](n≥1); and the extension ring F[x1,…,xn,z]/(zD−x1⋯xn) of F[x1,…,xn].