On the Geometry of Some Special Projective Varieties

Russo, Francesco

doi:10.1007/978-3-319-26765-4

Cited by 67 publications

(75 citation statements)

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“…If hess f = 0, then A has the SLP and the result follows. We recall that hypersurfaces of degree 4 with vanishing Hessian are classified by Gordan Noether Theorem (see [GN, CRS, GR] or [Ru,Chapter 7]). If hess f = 0, by Theorem [DP, Thm.…”

Section: 2mentioning

confidence: 99%

“…The Lefschetz properties for Artinian K-algebras are algebraic abstractions inspired by the Hard Lefschetz theorem on the cohomology of smooth projective varieties (see [La] or [Ru,Chapter 7]). Nowadays there are lots of contexts where the Lefschetz properties have been introduced, for instance Kahler manifolds, convex polytopes, Coxeter groups and tropical varieties (see [Ka,Be,HL,KN,NW,St,St2,BN,GZ]).…”

Section: Introductionmentioning

confidence: 99%

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The Jordan type of graded Artinian Gorenstein algebras

Costa

Gondim

2019

Advances in Applied Mathematics

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Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Jordan type of graded Artinian Gorenstein algebras

Costa

Gondim

2019

Advances in Applied Mathematics

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“…Using Proposition 2.3, Proposition 1.2.9 of either van den Essen (2000) or de Bondt (2009), and the above obtainedH (H ) = 0 and dim V (H ) ≤ n − 2, in that order, we deduce that…”

Section: Proposition 26mentioning

confidence: 58%

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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“…In this section we recall some basic facts about Artinian Gorenstein algebras and the Lefschetz properties. For a more detailed account, let see [HMMNWW,MN1,Ru,MW,Go].…”

Section: Artinian Gorenstein Algebras and The Lefschetz Propertiesmentioning

confidence: 99%

Lefschetz properties for higher order Nagata idealizations

Cerminara

Gondim

Ilardi

et al. 2019

Advances in Applied Mathematics

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We study a generalization of Nagata idealization for level algebras. These algebras are standard graded Artinian algebras whose Macaulay dual generator is given explicitly as a bigraded polynomial of bidegree (1, d). We consider the algebra associated to polynomials of the same type of bidegree (d 1 , d 2 ). We prove that the geometry of the Nagata hypersurface of order e is very similar to the geometry of the original hypersurface. We study the Lefschetz properties for Nagata idealizations of order d 1 , proving that WLP holds if d 1 ≥ d 2 . We give a complete description of the associated algebra in the monomial square free case.

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On the Geometry of Some Special Projective Varieties

Cited by 67 publications

References 0 publications

The Jordan type of graded Artinian Gorenstein algebras

The Jordan type of graded Artinian Gorenstein algebras

Homogeneous quasi-translations in dimension 5

Lefschetz properties for higher order Nagata idealizations

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