Symbolic powers of codimension two Cohen-Macaulay ideals

摘要

LetI(X)be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay schemeand letdenote its m-th symbolic power. We are interested in whenWe survey what is known about this problem when X is locally a complete intersection, and in particular, we review the classification of whenfor allWe 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 Romer.