Symbolic powers of codimension two Cohen-Macaulay ideals
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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.