# Symbolic powers of codimension two Cohen-Macaulay ideals Academic Article

•
• Overview
•
• Research
•
• Identity
•
• Additional Document Info
•
• View All
•

### 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.

### authors

• Van Tuyl, Adam
• Cooper, Susan
• Fatabbi, Giuliana
• Guardo, Elena
• Lorenzini, Anna
• Migliore, Juan
• Nagel, Uwe
• Seceleanu, Alexandra
• Szpond, Justyna
• Tuyl, Adam Van

### publication date

• November 1, 2020