Quasi Complete Intersections, Powers of Ideals, and Deficiency Modules

Peterson, Christopher S.

doi:10.1006/jabr.1997.7362

Cited by 5 publications

(5 citation statements)

References 16 publications

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“…(ii) If C ⊆ P 3 is a curve that is locally a complete intersection and is an almost complete intersection, then Theorem 2.3 gives [36,Corollary 2.7].…”

Section: Background Resultsmentioning

confidence: 99%

“…In fact, the original motivation of this project was to prove [17,Conjecture 4.1]. We were initially able to verify this conjecture using Peterson's [36] results on quasi-complete intersections, which first suggested the importance of being locally a complete intersection. Generalizing our specialized proof lead to the much stronger results of this paper.…”

Section: Introductionmentioning

confidence: 99%

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Symbolic powers of codimension two Cohen-Macaulay ideals

Cooper¹,

Fatabbi²,

Guardo³

et al. 2016

Preprint

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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.

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“…(ii) If C ⊆ P 3 is a curve that is locally a complete intersection and is an almost complete intersection, then Theorem 2.3 gives [36,Corollary 2.7].…”

Section: Background Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Symbolic powers of codimension two Cohen-Macaulay ideals

Cooper¹,

Fatabbi²,

Guardo³

et al. 2016

Preprint

View full text Add to dashboard Cite

show abstract

“…Proof. To show that (a) implies (b), we first note that an ACI set of points X is ACM by [22,Lemma 2.5], since X is obviously a local complete intersection. The claimed shape of α X follows then from [14,Cor.…”

Section: Now We Can Apply the Results Of The Previous Sections To Com...mentioning

confidence: 99%

Kaehler differentials for fat point schemes in P^1xP^1

Guardo,

Kreuzer,

Linh

et al. 2016

Preprint

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Let X be a set of K-rational points in P 1 × P 1 over a field K of characteristic zero, let Y be a fat point scheme supported at X, and let R Y be the bihomogeneus coordinate ring of Y. In this paper we investigate the module of Kähler differentials Ω 1 R Y /K . We describe this bigraded R Y -module explicitly via a homogeneous short exact sequence and compute its Hilbert function in a number of special cases, in particular when the support X is a complete intersection or an almost complete intersection in P 1 × P 1 . Moreover, we introduce a Kähler different for Y and use it to characterize ACM reduced schemes in P 1 × P 1 having the Cayley-Bacharach property.

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“…In fact, the original motivation of this project was to prove [18,Conjecture 4.1]. We were initially able to verify this conjecture using Peterson's [35] results on quasi-complete intersections, which first suggested the importance of being locally a complete intersection. Generalizing our specialized proof lead to the much stronger results of this paper.…”

Section: Introductionmentioning

confidence: 99%

“…2 unless X is a complete intersection. (ii) If C P 3 is a curve that is locally a complete intersection and is an almost complete intersection, then Theorem 2.3 gives[35, Corollary 2.7]. (iii) A connection between the number of generators and the equality of the regular and symbolic powers in a special case can be found in[28, Corollary 2.5].…”

mentioning

confidence: 99%

Symbolic powers of codimension two Cohen-Macaulay ideals

Cooper

Fatabbi²,

Guardo³

et al. 2020

Communications in Algebra

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Let I X be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme X P n , and let I ðmÞ X denote its m-th symbolic power. We are interested in when I ðmÞ X ¼ I m X : 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 ðmÞ X ¼ I m X 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 Â P 1 ; (2) verify a conjecture of Guardo, Harbourne, and Van Tuyl, and (3) provide additional evidence to a conjecture of R€ omer.

show abstract

Quasi Complete Intersections, Powers of Ideals, and Deficiency Modules

Cited by 5 publications

References 16 publications

Symbolic powers of codimension two Cohen-Macaulay ideals

Symbolic powers of codimension two Cohen-Macaulay ideals

Kaehler differentials for fat point schemes in P^1xP^1

Symbolic powers of codimension two Cohen-Macaulay ideals

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