2016/06/03 by Cooper, Susan, Fatabbi, Giuliana, Guardo, Elena +6
#Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1606.00935
Let IX be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme X ⊆ ℙn, and let IX(m) denote its m-th symbolic power. We are interested in when IX(m) = IXm. We survey what is known about this problem when X is locally a complete intersection, and in particular, we review the classification of when IX(m) = IXm 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 ℙ1 × ℙ1; (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of Römer.