Symbolic powers of codimension two Cohen-Macaulay ideals

Abstract

Let I X be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme X ⊆ P n , and let I (m) X denote 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 whenX 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.

Extreme values of the resurgence for homogeneous ideals in polynomial rings

“…If I(Z) is a complete intersection, then sdefect(I(Z), m) = 0 for all m ≥ 1 since we have I(mZ) = I(Z) m (indeed, if I is generated by a regular sequence, then I r = I (r) for all r ≥ 0 by [ZS,Lemma 5,Appendix 6]). When N = 2 and Z is reduced (i.e., I(Z) is radical), [GGSV,Theorem 2.6] gives a converse: if sdefect(I(Z), m) = 0 for all m ≥ 1, then I(Z) is a complete intersection (see also [CFGLMNSSV,Remark 2.5] and [HU,Theorem 2.8]).…”

The symbolic defect of an ideal

“…Weyman's paper [42] gives the resolution of Sym 2 (I). As shown in [12,39], the hypotheses on I imply that Sym 2 (I) ∼ = I 2 . Many of our arguments make use of Hilbert functions.…”

“…Let I be a homogeneous radical ideal of R. Recall that I is a generic complete intersection if the localization of I at any minimal associated prime of I is a complete intersection. A result of [12,39,42] will prove useful: [39][42]). Let I be a homogeneous ideal of k[x 0 , .…”

“…Note that in this case the linear star configuration defines a collection of points in P 2 . By applying [26, Corollary 3.9] of Harbourne and Huneke (and see also [12,Example 3.9] for additional details), we have the following relationship between the regular and symbolic powers of I 2,L in P 2 .…”

Fattening of ACM arrangements of codimension 2 subspaces in ℙN

“…In particular, Janssen in [13] generalized results of Bocci and Chiantini to configurations of lines in P 3 defined by homogeneous Cohen-Macaulay ideals. Symbolic powers of codimension 2 Cohen-Macaulay ideals have been studied recently in [4]. Results of these two articles, especially Section 3 in [13], have motivated our research presented here.…”