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 ) ⊆ I r for all r > 0 and all homogeneous ideals I in S = F [ P N ] = F [ x 0 , … , x N ] . The conjecture has been disproven for select values of N ≥ 2 : first by Dumnicki, Szemberg, and Tutaj-Gasinska 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 ) ⊆ I r 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 [ x 1 , … , x n ] ( n ≥ 1 ) ; and the extension ring F [ x 1 , … , x n , z ] / ( z D − x 1 ⋯ x n ) of F [ x 1 , … , x n ] .