2016
DOI: 10.48550/arxiv.1606.00935
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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 al… Show more

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Cited by 4 publications
(7 citation statements)
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References 31 publications
(61 reference statements)
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“…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]).…”
Section: Resultsmentioning
confidence: 99%
“…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]).…”
Section: Resultsmentioning
confidence: 99%
“…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.…”
Section: Remark 21mentioning
confidence: 91%
“…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 , .…”
Section: Remark 21mentioning
confidence: 99%
See 1 more Smart Citation
“…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.…”
Section: Introductionmentioning
confidence: 99%