Logarithmic vector fields along smooth divisors in projective spaces

Ueda, Kanji; Yoshinaga, Masahiko

doi:10.14492/hokmj/1258553970

Cited by 21 publications

(54 citation statements)

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“…In the next section we recall the main results concerning this problem in the case of hyperplanes ( [11], [26], [1]), of one smooth hypersurface ( [25], [24], [2]), of many smooth hypersurfaces of degree d ≥ 2 and of two smooth quadrics ( [2]). In some of them, the components of D are recovered by looking at the set of unstable objects of Ω 1 P n (log D) of a given degree; to that end we make the following: Definition 2.4.…”

Section: Preliminary Definitions and Notationsmentioning

confidence: 99%

Logarithmic bundles of hypersurface arrangements in $$\mathbf{P}^n$$ P n

Angelini

2014

Collect. Math.

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Let D = {D1, . . . , D } be a multi-degree arrangement with normal crossings on the complex projective space P n , with degrees d1, . . . , d ; let Ω 1 P n (log D) be the logarithmic bundle attached to it. First we prove a Torelli type theorem when D has a sufficiently large number of components by recovering them as unstable smooth irreducible degree-di hypersurfaces of Ω 1 P n (log D). Then, when n = 2, by describing the moduli spaces containing Ω 1 P 2 (log D), we show that arrangements of a line and a conic, or of two lines and a conic, are not Torelli. Moreover we prove that the logarithmic bundle of three lines and a conic is related with the one of a cubic. Finally we analyze the conic-case.

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Section: Preliminary Definitions and Notationsmentioning

confidence: 99%

Logarithmic bundles of hypersurface arrangements in $$\mathbf{P}^n$$ P n

Angelini

2014

Collect. Math.

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“…Recall the following notion. For a discussion of this notion and various examples we refer to [14], [28], [13]. In particular, E. Sernesi and the author have shown in [13] that the nodal curves with a small number of nodes are DK-Torelli.…”

Section: Introductionmentioning

confidence: 99%

Jacobian syzygies, stable reflexive sheaves, and Torelli properties for projective hypersurfaces with isolated singularities

Dimca¹

2017

Alg. Geom.

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We investigate the relations between the syzygies of the Jacobian ideal of the defining equation for a projective hypersurface V with isolated singularities and the Torelli properties of V (in the sense of Dolgachev-Kapranov). We show, in particular, that hypersurfaces with a small Tjurina number are Torelli in this sense. When V is a plane curve or, more interestingly, a surface in P 3 , we discuss the stability of the reflexive sheaf of logarithmic vector fields along V . A new lower bound for the minimal degree of a syzygy associated with a 1-dimensional almost complete intersection is also given.

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“…In a 1983 paper [, § 4], Max Benson proved this result on the locus of smooth hypersurfaces. This was subsequently rediscovered several times, for example, in [, Lemma 3; , Corollary 1.3]. In a more recent paper[, Theorem 1.1], Wang proved that if

\nabla

fails to be injective at

\overset{f}{true}

, then either

f

is a direct sum or

f

has a point of multiplicity

deg (f) - 1

.…”

Section: Direct Sums and Non‐injectivity Of ∇mentioning

confidence: 99%

“…16 (see[4, Theorem 1.7]). A concise form f ∈ S d+1 is either a direct sum or an LDS form if and only if the apolar ideal f ⊥ has a minimal generator in degree d + 1.…”

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confidence: 99%

Direct sum decomposability of polynomials and factorization of associated forms

Fedorchuk

2020

Proc. London Math. Soc.

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We prove two criteria for direct sum decomposability of homogeneous polynomials. For a homogeneous polynomial with a non‐zero discriminant, we interpret direct sum decomposability of the polynomial in terms of factorization properties of the Macaulay inverse system of its Milnor algebra. This leads to an if‐and‐only‐if criterion for direct sum decomposability of such a polynomial, and to an algorithm for computing direct sum decompositions over any field, either of characteristic 0 or of sufficiently large positive characteristic, for which polynomial factorization algorithms exist. For homogeneous forms over algebraically closed fields, we interpret direct sums and their limits as forms that cannot be reconstructed from their Jacobian ideal. We also give simple necessary criteria for direct sum decomposability of arbitrary homogeneous polynomials over arbitrary fields and apply them to prove that many interesting classes of homogeneous polynomials are not direct sums.

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Logarithmic vector fields along smooth divisors in projective spaces

Abstract: We show that a smooth divisor in a projective space can be reconstructed from the isomorphism class of the sheaf of logarithmic vector fields along it if and only if its defining equation is of Sebastiani-Thom type.

Cited by 21 publications

References 1 publication

Logarithmic bundles of hypersurface arrangements in $$\mathbf{P}^n$$ P n

Logarithmic bundles of hypersurface arrangements in $$\mathbf{P}^n$$ P n

Jacobian syzygies, stable reflexive sheaves, and Torelli properties for projective hypersurfaces with isolated singularities

Direct sum decomposability of polynomials and factorization of associated forms

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