Kähler differentials and Kähler differents for fat point schemes

Kreuzer, Martin; Linh, Tran N. K.; Long, Le Ngoc

doi:10.1016/j.jpaa.2015.02.028

Cited by 12 publications

(9 citation statements)

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“…In the proof of Lemma 3.2(b), there exist k, l ∈ N such that h := x k 0 y l 0 ∈ ϑ X . In particular, we may choose k = (m + 1)(s 1 − 1) and l = (n + 1)(s 2 − 1) by [11,Proposition 3.5]. So, the multiplication map (R X ) i,j ×h → (ϑ X ) (i+k,j+l) is injective as K-vector spaces.…”

Section: The Kähler Differentmentioning

confidence: 99%

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The Kähler Different of a Set of Points in $\mathbb{P}^m\times\mathbb{P}^n$

Linh,

Long,

Hoa

et al. 2021

Preprint

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Given an ACM set X of points in a multiprojective space P m × P n over a field of characteristic zero, we are interested in studying the Kähler different and the Cayley-Bacharach property for X. In P 1 × P 1 , the Cayley-Bacharach property agrees with the complete intersection property and it is characterized by using the Kähler different. However, this result fails to hold in P m × P n for n > 1 or m > 1. In this paper we start an investigation of the Kähler different and its Hilbert function and then prove that X is a complete intersection of type (d 1 , ..., dm, d 1 , ..., d n ) if and only if it has the Cayley-Bachrach property and the Kähler different is non-zero at a certain degree. When X has the ( )-property, we characterize the Cayley-Bacharach property of X in terms of its components under the canonical projections.

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Section: The Kähler Differentmentioning

confidence: 99%

“…Let F i ∈ K[X 0 , ..., X m ] be a minimal separator of q i in X 1 and F 1 ∈ K[Y 0 , ..., Y n ] be a minimal separator of q 1 in X 2 . By [11,Corollary 2.6], the image of F m i in R X1 belongs to ϑ X1 and the image of F n 1 in R X2 belongs to ϑ X2 . So, the image of…”

Section: Special Acm Setsmentioning

confidence: 99%

The Kähler Different of a Set of Points in $\mathbb{P}^m\times\mathbb{P}^n$

Linh,

Long,

Hoa

et al. 2021

Preprint

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“…In [5], De Dominicis and the first author initiated a careful examination of the structure and the Hilbert function of Ω 1 R X /K for a set of points X (i.e., for the case m 1 = • • • = m s = 1). Then, in [8], the present authors started the study of the modules of Kähler differential k-forms for fat point schemes, and in [9] this work was continued.…”

Section: Introductionmentioning

confidence: 99%

“…The research underlying this paper and the calculation of the examples were greatly aided by an implementation of the relevant objects and functions in the computer algebra system ApCoCoA (see [2]). Unless explicitly stated otherwise, we adhere to the definitions and notation introduced in the books [10] and [11], as well as in our previous papers [8] and [9].…”

Section: Introductionmentioning

confidence: 99%

Hilbert Polynomials of Kähler Differential Modules for Fat Point Schemes

Kreuzer¹,

Linh²,

Long³

2020

Preprint

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Given a fat point scheme W = m 1 P 1 + • • • + m s P s in the projective n-space P n over a field K of characteristic zero, the modules of Kähler differential k-forms of its homogeneous coordinate ring contain useful information about algebraic and geometric properties of W when k ∈ {1, . . . , n + 1}. In this paper we determine the value of its Hilbert polynomial explicitly for the case k = n + 1, confirming an earlier conjecture. More precisely this value is given by the multiplicity of the fat point scheme Y = (m 1 −1)P 1 +• • •+(m s −1)P s . For n = 2, this allows us to determine the Hilbert polynomials of the modules of Kähler differential k-forms for k = 1, 2, 3, and to produce a sharp bound for the regularity index for k = 2.

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“…, d n then it follows that the Hilbert function of Ω 1 R X /K is given by HF Ω 1 R X /K (i) = (n + 1) HF X (i − 1) − n j=1 HF X (i − d j ) for all i ∈ Z. Later, in [2,4], the differential algebra techniques were extended to fat point schemes in P n and in P 1 × P 1 .…”

Section: Introductionmentioning

confidence: 99%