Symbolic powers of codimension two Cohen-Macaulay ideals

Let IX be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme and let denote its m-th symbolic power. We are interested in when We survey what is known about this problem when X is locally a complete intersection, and in particular, we review the classification of when for all 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 (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of Römer.