The Aluffi algebra of the Jacobian of points in projective space: Torsion-freeness

Nejad, Abbas Nasrollah; Simis, Aron; Zaare-Nahandi, Rashid

doi:10.1016/j.jalgebra.2016.08.012

Cited by 5 publications

(6 citation statements)

References 10 publications

(16 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The same argument as in Proposition 3.3 implies that I 2 (Θ) = (x, y, z) 4 , where Θ is the Jacobian matrix of I(X). Therefore, the assertion follows by [9,Lemma 1.4].…”

Section: Letmentioning

confidence: 83%

“…Since I 2 (Θ) ⊆ (x, y, z) 4 , it follows that I 2 (Θ) = (x, y, z) 4 . The second assertion follows by [9,Lemma 1.4].…”

mentioning

confidence: 85%

“…The results and examples in [9] show that the answer to this problem may not have a general shape, even when one assumes that the defining ideal is generated by quadrics -a condition that is only guaranteed when the number of points in P n is at most 2n. In P 2 this forces the number of points to be at most 4, not a very bright situation.…”

Section: Introductionmentioning

confidence: 99%

“…In [9] the problem was circumscribed to the case of a set of reduced points when the latter admits a minimal set of generators in degree 2. In [7,Theorem 1.2] a characterization of the vanishing of the module of Valabrega-Valla was given in terms of the first syzygy module of the form ideal J * in the associated graded ring of I.…”

Section: Introductionmentioning

confidence: 99%

“…Since R/I(X) is locally regular on the punctured spectrum, by the Jacobian criterion it translates into the property that the Jacobian ideal I = (I(X), I n (Θ)) is m-primary, where I n (Θ) is the ideal of n-minors of the Jacobian matrix Θ of I(X) and m denotes the maximal irrelevant ideal of the polynomial ring R. In other words, there is a suitable power m l that lands into I. Therefore, if the defining ideal of X is minimally generated in single degree m ≥ 2 and m n(m−1) ⊂ I, then the Valabrega-Valla module vanishes [9].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The module of Valabrega-Valla of the Jacobian ideal of points in projective plane

Nejad,

Shahidi

2019

Preprint

Self Cite

View full text Add to dashboard Cite

The module of Valabrega-Valla of the Jacobian ideal of a reduced projective variety V is the torsion of the Aluffi algebra. One considers the problem of its vanishing in the case of where V is a reduced set of points in the projective plane. It is shown that the module is nonzero for several cases of a special configuration class therein -called (s − r)-fold collinear configuration. A complete classification of types is given for 5 and 6 points in regard to this problem.

show abstract

“…The same argument as in Proposition 3.3 implies that I 2 (Θ) = (x, y, z) 4 , where Θ is the Jacobian matrix of I(X). Therefore, the assertion follows by [9,Lemma 1.4].…”

Section: Letmentioning

confidence: 83%

“…Since I 2 (Θ) ⊆ (x, y, z) 4 , it follows that I 2 (Θ) = (x, y, z) 4 . The second assertion follows by [9,Lemma 1.4].…”

mentioning

confidence: 85%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The module of Valabrega-Valla of the Jacobian ideal of points in projective plane

Nejad,

Shahidi

2019

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Gimenez

Martinez-Bernal

Simis

et al. 2018

Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

Self Cite

View full text Add to dashboard Cite

In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertexweighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen-Macaulay vertexweighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen-Macaulay and satisfy Alexander duality.

show abstract

Torsion-free Aluffi algebras

2018

Self Cite

View full text Add to dashboard Cite

A pair of ideals J ⊆ I ⊆ R has been called Aluffi torsion-free if the Aluffi algebra of I/J is isomorphic with the corresponding Rees algebra. We give necessary and sufficient conditions for the Aluffi torsion-free property in terms of the first syzygy module of the form ideal J * in the associated graded ring of I. For two pairs of ideals J 1 , J 2 ⊆ I such that J 1 − J 2 ∈ I 2 , we prove that if one pair is Aluffi torsion-free the other one is so if and only if the first syzygy modules of J 1 and J 2 have the same form ideals. We introduce the notion of strongly Aluffi torsion-free ideals and present some results on these ideals. t 2010 Mathematics Subject Classification. primary 13A30, 13C12, 14C17; secondary 14B05, 13E15. 1 2 A. NASROLLAH NEJAD, Z. SHAHIDI, R. ZAARE-NAHANDIgenerate J * is an essential problem in resolution of singularities. Also the torsionfree Aluffi algebras are crucial in intersection theory of regular and linear embedding ([4], [10]). The outline of this paper is as the follow.In section 1, we give necessary and sufficient conditions for torsion-free Aluffi algebra, involving the standard base (in the sense of Hironaka ([7])) and the first syzygy module of the form ideal in the associated graded ring.Let J ⊆ I be ideals in the ring R. We say that the pair J ⊆ I is Aluffi torsion-free if J ∩ I n = JI n−1 for all n ≥ 1. In section 2, we study the behavior of the Aluffi torsion-free property with respect to contraction and extension. We prove that the sum of two Aluffi torsion-free ideals is Aluffi torsion-free if and only if one of them modulo the other is Aluffi torsion-free. As the main result of this section, we prove that if J 1 , J 2 ⊆ I such that J 1 ≡ J 2 modulo I 2 and J 1 ⊆ I is Aluffi torsion-free then J 2 ⊆ I is Aluffi torsion-free if and only if the first syzygy modules of J 1 and J 2 have the same form ideals in the associated graded module gr I (R m ) where m is the number of generators of J 1 and J 2 (Theorem 2.6). In sequel, we introduce the notion of strongly Aluffi torsion-free ideals. A pair J = (f 1 , . . . , f t ) ⊆ I ⊆ R is called strongly Aluffi torsion-free if J i = (f 1 , . . . , f i ) is Aluffi torsion-free for i = 1, . . . , t. We give an example of Aluffi torsion-free pair of ideals which is not strongly Aluffi torsion-free. In the case that, J ⊆ I is Aluffi torsion-free, we give a criterion for strongly Aluffi torsion-freeness. We close the section with this result: let J 1 , J 2 ⊆ I be ideals in the ring R such that the extension of J 2 and I in the ring R/J 1 is Aluffi torsion-free. If there exists a minimal generating set f 1 , . . . , f t of J 1 such that J 1 ⊆ I is strongly Aluffi torsion-free and extension of the sequence f 1 , . . . , f t in R/J 2 is regular then J 2 ⊆ I is Aluffi torsion-free.In section 3, we focus on the case that J is an ideal in the polynomial ring R = k[x 0 , . . . , x n ] over a field k of characteristic zero and the ideal I stands for the Jacobian ideal of J which describe the singular subscheme of Spec (R/J). We prove that if J is...

show abstract

The Aluffi algebra of the Jacobian of points in projective space: Torsion-freeness

Cited by 5 publications

References 10 publications

The module of Valabrega-Valla of the Jacobian ideal of points in projective plane

The module of Valabrega-Valla of the Jacobian ideal of points in projective plane

Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

Torsion-free Aluffi algebras

Contact Info

Product

Resources

About