A formula for symbolic powers

TL;DR

This paper develops new formulas and criteria for understanding symbolic powers of ideals in Cohen-Macaulay rings, including conditions for equality with ordinary powers and bounds on their degrees, advancing algebraic ideal theory.

Contribution

It introduces a multiplicity-based characterization of symbolic power equality, a saturation formula for symbolic powers, and bounds on initial degrees, also proving a conjecture by Eisenbud and Mazur.

Findings

Characterized when $J=I^{(m)}$ using multiplicity.

Derived a saturation formula for $I^{(m)}$.

Proved a conjecture on ${ m ann}_S(I^{(m)}/I^m)$.

Abstract

Let $S$ be a Cohen-Macaulay ring which is local or standard graded over a field, and let $I$ be an unmixed ideal that is also generically a complete intersection. Our goal in this paper is multi-fold. First, we give a multiplicity-based characterization of when an unmixed subideal $J \subseteq I^{(m)}$ equals the $m$-th symbolic power $I^{(m)}$ of $I$. Second, we provide a saturation-type formula to compute $I^{(m)}$ and employ it to deduce a theoretical criterion for when $I^{(m)}=I^m$. Third, we establish an explicit linear bound on the exponent that makes the saturation formula effective, and use it to obtain lower bounds for the initial degree of $I^{(m)}$. Along the way, we prove a conjecture (in fact, a generalized version of it) due to Eisenbud and Mazur about ${\rm ann}_S(I^{(m)}/I^m)$, and we propose a conjecture connecting the symbolic defect of an ideal to Jacobian ideals.