A Geometrical approach to Gordan-Noether's and Franchetta's contributions to a question posed by Hesse

Garbagnati, Alice; Repetto, Flavia

doi:10.1007/bf03191214

Cited by 22 publications

(17 citation statements)

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“…The aim of this appendix is to recall classical results and constructions of hypersurfaces with vanishing hessian whose original work was writen in German (see [GN]) and Italian (see [Pe,Pt1,Pt2,Pt3]). Some of this classical work was revisited in [CRS,GR,Wa1,Lo,GRu,Ru,dB].…”

Section: Appendix: Forms With Vanishing Hessian An Overviewmentioning

confidence: 99%

On higher Hessians and the Lefschetz properties

Gondim

2017

Journal of Algebra

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We deal with a generalization of a Theorem of P. Gordan and M. Noether on hypersurfaces with vanishing (first) Hessian. We prove that for any given N ≥ 3, d ≥ 3 and 2 ≤ k < d 2 there are irreducible hypersurfaces X = V (f ) ⊂ P N , of degree deg(f ) = d, not cones and such that their Hessian of order k, hess k f , vanishes identically. The vanishing of higher Hessians is closely related with the Strong (or Weak) Lefschetz property for standard graded Artinian Gorenstein algebra, as pointed out first in [Wa1] and later in [MW]. As an application we construct for each pair (N, d) = (3, 3), (3, 4), standard graded Artinian Gorenstein algebras A, of codimension N + 1 ≥ 4 and with socle degree d ≥ 3 which do not satisfy the Strong Lefschetz property, failing at an arbitrary step k with 2 ≤ k < d 2 . We also prove that for each pair (N, d), N ≥ 3 and d ≥ 3 except (3, 3), (3, 4), (3, 6) and (4, 4) there are standard graded Artinian Gorenstein algebras of codimension N + 1, socle degree d, with unimodal Hilbert vectors and which do not satisfy the Weak Lefschetz property.

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Section: Appendix: Forms With Vanishing Hessian An Overviewmentioning

confidence: 99%

On higher Hessians and the Lefschetz properties

Gondim

2017

Journal of Algebra

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“…4. This connection is used in Garbagnati and Repetto (2009) and Russo (2016, Ch. 7) as well, and appears as Garbagnati and Repetto (2009, p. 33) and Russo (2016, Lem.…”

Section: Introductionmentioning

confidence: 99%

“…7.3.7) respectively. Garbagnati and Repetto (2009) and Russo (2016, Ch. 7) contain classifications in dimensions less than 5 as well, but with the same limitations as above on the factorization of h. These limitations are not present in Lossen (2004), which follows the approach of Gordan and Nöther (1876) in proving the classifications in dimensions less than 5.…”

Section: Introductionmentioning

confidence: 99%

“…4. Garbagnati and Repetto (2009) and Russo (2016, Ch. 7) on one hand, and de Bondt (2009, Th. 5.3.7) on the other hand, treat the case where rk J H = 3 incorrectly as well.…”

Section: Introductionmentioning

confidence: 99%

“…5.3.7) fix each other's errors. The error in Garbagnati and Repetto (2009) and Russo (2016, Ch. 7) can be repaired by way of Theorem 4.6, which comes from Gordan and Nöther (1876).…”

Section: Introductionmentioning

confidence: 99%

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Homogeneous quasi-translations in dimension 5

Bondt

2017

Beitr Algebra Geom

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We give a proof in modern language of the following result by Paul Gordan and Max Nöther: a homogeneous quasi-translation in dimension 5 without linear invariants would be linearly conjugate to another such quasi-translation x + H , for which H 5 is algebraically independent over C of H 1 , H 2 , H 3 , H 4 . Just like Gordan and Nöther, we apply this result to classify all homogeneous polynomials h in 5 indeterminates, for which the Hessian determinant is zero. Others claim to have reproved 'the result of Gordan and Nöther in P 4 ' as well, but their proofs have gaps, which can be fixed by using the above result about homogeneous quasi-translations. Furthermore, some of the proofs assume that h is irreducible, which Gordan and Nöther did not. We derive some other properties which H would have. One of them is that deg H ≥ 15, for which we give a proof which is less computational than another proof of it by Dayan Liu. Furthermore, we show that the Zariski closure of the image of H would be an irreducible component of V (H ), and prove that every other irreducible component of V (H ) would be a 3-dimensional linear subspace of C 5 which contains the fifth standard basis unit vector.

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On the Theory of Gordan-Noether on Homogeneous Forms with Zero Hessian (Improved Version)

Watanabe¹,

Bondt²

2020

Springer Proceedings in Mathematics &Amp; Statistics

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We give a detailed proof for Gordan-Noether's results in "Ueber die algebraischen Formen, deren Hesse'sche Determinante identisch verschwindet." C. Lossen has written a paper in a similar direction as the present paper, but did not provide a proof for every result. In our paper, every result is proved. Furthermore, our paper is independent of Lossen's paper and includes a considerable number of new observations. An earlier version of this paper, namely [17], has been printed in version, a serious error has been corrected, namely [17, Lemma 5.2]. Lemma 5.2 has been replaced by a weaker statement, and Proposition 6.2 has been weakened along with that. Lemma 5.2 no longer suffices for the proof of Proposition 7.2. For that reason, a new section (Section 8) has been added to complete the proof of Proposition 7.2. Furthermore, several trivial errors have been corrected, and a section with new results has been added (Section 9).

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A Geometrical approach to Gordan-Noether's and Franchetta's contributions to a question posed by Hesse

Cited by 22 publications

References 9 publications

On higher Hessians and the Lefschetz properties

On higher Hessians and the Lefschetz properties

Homogeneous quasi-translations in dimension 5

On the Theory of Gordan-Noether on Homogeneous Forms with Zero Hessian (Improved Version)

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