Symbolic powers of codimension two Cohen-Macaulay ideals

TL;DR

This paper reviews conditions under which symbolic and ordinary powers of codimension two Cohen-Macaulay ideals coincide, explores weakening these conditions, and applies findings to specific geometric and algebraic conjectures.

Contribution

It surveys known classifications for when symbolic and ordinary powers are equal, discusses potential relaxations, and applies results to particular cases and conjectures.

Findings

Classification of when $I_X^{(m)} = I_X^m$ for all $m$

Simplification of symbolic powers for points in $P^1 imes P^1$

Verification of a conjecture by Guardo, Harbourne, and Van Tuyl

Abstract

Let $I_X$ be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme $X \subseteq \mathbb{P}^n$, and let $I_X^{(m)}$ denote its $m$-th symbolic power. We are interested in when $I_X^{(m)} = I_X^m$. We survey what is known about this problem when $X$ is locally a complete intersection, and in particular, we review the classification of when $I_X^{(m)} = I_X^m$ for all $m \geq 1$. We then discuss how one might weaken these hypotheses, but still obtain equality between the symbolic and ordinary powers. Finally, we show that this classification allows one to: (1) simplify known results about symbolic powers of ideals of points in $\mathbb{P}^1 \times \mathbb{P}^1$; (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of R\"omer.