Symbolic powers of codimension two Cohen-Macaulay ideals. Issue 11 (1st November 2020) - Cooper, Susan; Fatabbi, Giuliana; Guardo, Elena; Lorenzini, Anna; Migliore, Juan; Nagel, Uwe; Seceleanu, Alexandra; Szpond, Justyna; Tuyl, Adam Van
Abstract: Let IX be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme X ⊆ 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 ≥ 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 P 1 × P 1 ; (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of Römer.