The equations of Rees algebras of equimultiple ideals of deviation one

Muiños, Ferran; Planas-Vilanova, Francesc

doi:10.1090/s0002-9939-2012-11398-x

Cited by 7 publications

(6 citation statements)

References 31 publications

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“…W. Heinzer and M. -K. Kim proved that if I is an equimultiple ideal of deviation 1 and depth(G(I )) ≥ grade(I ) − 1, then reltype(F (I )) = reltype(R(I )) = r +1, where r is the reduction number of I (see [7]). In [12], F. Muiños and F. Planas-Vilanova recover this result by proving in addition that there is a unique defining equation of R(I ) of maximal degree. Also, their work recovers Theorem 2.3.3 in [19], for the case reduction number equal to 1 (see also [9]).…”

Section: Introductionmentioning

confidence: 75%

“…we get depth(G(F m )) ≥ depth(G(I )) ≥ depth(G(F m )) − 1. Therefore, the assumption of Theorem 3.5 is weaker than the assumptions of the main Theorem of [12]. In addition, suppose depth(G(F m )) ≥ n − 1.…”

Section: Equimultiple Ideals Of Deviation Onementioning

confidence: 99%

“…We begin the section by giving a simple proof of the main theorem of Muiños and Planas-Vilanova [12].…”

Section: Equimultiple Ideals Of Deviation Onementioning

confidence: 99%

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On the Relation Type of Fiber Cone

Jayanthan

Nanduri

2015

Acta Math Vietnam

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In this article, we study the relation type of the fiber cone of certain special classes of ideals in Noetherian local rings. We show that in any Noetherian local ring, if deviation of I is 1, and depth(G(F L )) ≥ − 1, then the relation types of R(I ) and F L (I ) are equal. We also prove that for lexsegment ideals in K [x, y], where K is a field, the relation types of the fiber cone and the Rees algebra are equal.

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Section: Introductionmentioning

confidence: 75%

Section: Equimultiple Ideals Of Deviation Onementioning

confidence: 99%

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On the Relation Type of Fiber Cone

Jayanthan

Nanduri

2015

Acta Math Vietnam

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“…Remark 2.15. (a) When (R, m) is a regular local ring and I is an m-primary ideal whcih is an almost complete intersection, there is an a priori precise relation between the reduction number with respect to reduction generated by a regular sequence and the relation type of I (see [10]). Since we are dealing with forms of different degrees, we felt safer to give an independent self-contained argument that does not assume that the relation type is ≤ 3.…”

Section: Second Step (Under the Assumptions Of The Last Item In The mentioning

confidence: 99%

The ubiquity of Sylvester forms in almost complete intersections

Simis

Tohǎneanu

2014

Collect. Math.

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We study the structure of the Rees algebra of almost complete intersection ideals of finite colength in low-dimensional polynomial rings over fields. The main tool is a mix of Sylvester forms and iterative mapping cone construction. The material developed spins around ideals of forms in two or three variables in the search of those classes for which the corresponding Rees ideal is generated by Sylvester forms and is almost Cohen-Macaulay. A main offshoot is in the case where the forms are monomials. Another consequence is a proof that the Rees ideals of the base ideals of certain plane Cremona maps (e.g., de Jonquières maps) are generated by Sylvester forms and are almost Cohen-Macaulay.

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“…In [5] Heinzer and Kim proved that the equation of the fiber cone of an ideal is generated by a single equation of degree. In [8] authors described the equation of the Rees algebra of an equimultiple I of deviation and proved that there is a unique equation of maximum degree in a minimal generating set of defining equation of the Rees algebra.…”

Section: Introductionmentioning

confidence: 99%