Infinitesimal Variation Functions for Families of Smooth Varieties

Favale, Filippo F.; Pirola, Gian Pietro

doi:10.1007/s00032-022-00353-2

Cited by 3 publications

(4 citation statements)

References 21 publications

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“…In [BFP22] and in [FP22] the authors have developped a technical lemma which gives strong informations on the locus Θ introduced above. We present now a generalised version of it.…”

Section: Notations Preliminaries and Technical Resultsmentioning

confidence: 99%

“…In this article we exploit and generalize some techniques that the authors have developed in previous works on these topics, in collaboration with Gian Pietro Pirola (see [FP22] and [BFP22]) in order to prove the validity of some Lefschetz properties for particular Artinian algebras. More precisely, we deal with standard Artinian Gorenstein graded algebras over a field K of characteristic 0.…”

Section: Introductionmentioning

confidence: 99%

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Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

Bricalli¹,

Favale²

2022

Preprint

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In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree 1 for a complete intersection standard Artinian Gorenstein algebra of codimension 6 presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of "special" Artinian algebras: for them we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree 1 and the "lifting" of a weak Lefschetz property to an algebra from one of its quotients.

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“…In [BFP22] and in [FP22] the authors have developped a technical lemma which gives strong informations on the locus Θ introduced above. We present now a generalised version of it.…”

Section: Notations Preliminaries and Technical Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

Bricalli¹,

Favale²

2022

Preprint

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“…In [4] and in [9] the authors have developped a technical lemma which gives strong informations on the locus introduced above. We present now a generalised version of it.…”

Section: Construction 15 To Summarize If R Is a Saga Of Codimension N...mentioning

confidence: 99%

“…In this article we exploit and generalize some techniques that the authors have developed in previous works on these topics, in collaboration with Gian Pietro Pirola (see [9] and [4]) in order to prove the validity of some Lefschetz properties for particular Artinian algebras. More precisely, we deal with standard Artinian Gorenstein graded algebras over a field K of characteristic 0.…”

Section: Introductionmentioning

confidence: 99%

Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

Bricalli

Favale

2022

Collect. Math.

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In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree 1 for a complete intersection standard Artinian Gorenstein algebra of codimension 6 presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of “special” Artinian algebras: for them we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree 1 and the “lifting” of a weak Lefschetz property to an algebra from one of its quotients.

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A theorem of Gordan and Noether via Gorenstein rings

Bricalli,

Favale,

Pirola

2023

Sel. Math. New Ser.

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Gordan and Noether proved in their fundamental theorem that an hypersurface $$X=V(F)\subseteq {{\mathbb {P}}}^n$$ X = V ( F ) ⊆ P n with $$n\le 3$$ n ≤ 3 is a cone if and only if F has vanishing hessian (i.e. the determinant of the Hessian matrix). They also showed that the statement is false if $$n\ge 4$$ n ≥ 4 , by giving some counterexamples. Since their proof, several others have been proposed in the literature. In this paper we give a new one by using a different perspective which involves the study of standard Artinian Gorenstein $${{\mathbb {K}}}$$ K -algebras and the Lefschetz properties. As a further application of our setting, we prove that a standard Artinian Gorenstein algebra $$R={{\mathbb {K}}}[x_0,\dots ,x_4]/J$$ R = K [ x 0 , ⋯ , x 4 ] / J with J generated by a regular sequence of quadrics has the strong Lefschetz property. In particular, this holds for Jacobian rings associated to smooth cubic threefolds.

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Infinitesimal Variation Functions for Families of Smooth Varieties

Cited by 3 publications

References 21 publications

Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

A theorem of Gordan and Noether via Gorenstein rings

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