Uniform Harbourne-Huneke Bounds via Flat Extensions

TL;DR

This paper investigates conditions under which uniform Harbourne-Huneke bounds hold for certain classes of ideals, especially monomial primes, in normal rings, providing effective criteria and bounds for specific algebraic structures.

Contribution

It establishes criteria for ideal containments of the form I^{(N(r-1)+1)} subseteq I^r in normal rings, extending known results to new classes like monomial primes in tensor products.

Findings

Containment criteria for various classes of ideals in normal rings.

Effective bounds for monomial primes in tensor products of affine semigroup rings.

Extension of Harbourne conjecture bounds to specific algebraic structures.

Abstract

Over an arbitrary field $\mathbb{F}$, Harbourne conjectured that $$I^{(N (r-1)+1)} \subseteq I^r$$ for all $r>0$ and all homogeneous ideals $I$ in $S = \mathbb{F} [\mathbb{P}^N] = \mathbb{F} [x_0, \ldots, x_N]$. The conjecture has been disproven for select values of $N \ge 2$: first by Dumnicki, Szemberg, and Tutaj-Gasi\'{n}ska in characteristic zero, and then by Harbourne and Seceleanu in odd positive characteristic. However, the ideal containments above do hold when, for instance, $I$ is a monomial ideal in $S$. As a sequel to (arXiv:1510.02993), we present criteria for containments of type $I^{(N (r-1)+1)} \subseteq I^r$ 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 $\mathbb{F} [x_1, \ldots, x_n]$ $(n \ge 1)$; and the extension ring $\mathbb{F} [x_1, \ldots, x_n, z]/(z^D - x_1 \cdots x_n)$ of $\mathbb{F}[x_1, \ldots, x_n]$.