VSPs of cubic fourfolds and the Gorenstein locus of the Hilbert scheme of 14 points on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">A</mml:mi></mml:mrow><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msup></mml:math>

Jelisiejew, Joachim

doi:10.1016/j.laa.2018.08.002

Cited by 12 publications

(16 citation statements)

References 29 publications

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“…Remark 4.17. As follows from [25] there are non-smoothable closed subschemes of P 6 with Hilbert function (1, 7, 13, 14, 14 . .…”

Section: Thus It Is Enough To Prove Theorem 13 For Pointsmentioning

confidence: 99%

Identifying limits of ideals of points in the case of projective space

Mańdziuk

2020

Preprint

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In this paper we study the closure of the locus of radical ideals in the multigraded Hilbert scheme associated with a standard graded polynomial ring and the Hilbert function of a homogeneous coordinate ring of points in general position in projective space. In the case of projective plane, we give a sufficient condition for an ideal to be in the closure of the locus of radical ideals. For projective space of arbitrary dimension we present a necessary condition. The paper is motivated by the border apolarity lemma which connects such multigraded Hilbert schemes with the theory of ranks of polynomials.

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“…Remark 4.17. As follows from [25] there are non-smoothable closed subschemes of P 6 with Hilbert function (1, 7, 13, 14, 14 . .…”

Section: Thus It Is Enough To Prove Theorem 13 For Pointsmentioning

confidence: 99%

Identifying limits of ideals of points in the case of projective space

Mańdziuk

2020

Preprint

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“…The equations of

σ_{9} false(ν_{3} ({double-struckP}^{5}) false)

are already discussed in some papers. The invariant

T_{10}

corresponds to the Iliev–Ranestad divisor

D_{I R}

introduced in and studied by Ranestad and Voisin in [, § 2] and by Jelisiejew in [, Proposition 2]. Indeed it is observed in [, Remark 28] that

D_{I R}

is the

S L_{6}

‐invariant of smallest degree on

{Sym}^{3} C^{6}

.…”

Section: The Polynomials Sm⟨n⟩s and Sm⟨n⟩s0mentioning

confidence: 99%

“…The invariant

T_{10}

corresponds to the Iliev–Ranestad divisor

D_{I R}

introduced in and studied by Ranestad and Voisin in [, § 2] and by Jelisiejew in [, Proposition 2]. Indeed it is observed in [, Remark 28] that

D_{I R}

is the

S L_{6}

‐invariant of smallest degree on

{Sym}^{3} C^{6}

. Jelisiejew poses the interesting question if the degree 28 divisor of Theorem * is (up to multiples of

T_{10}

) the divisor

D_{V - a p}

of cubic fourfolds apolar to a Veronese surface .…”

Section: The Polynomials Sm⟨n⟩s and Sm⟨n⟩s0mentioning

confidence: 99%

Polynomials and the exponent of matrix multiplication

Chiantini

Hauenstein

Ikenmeyer

et al. 2018

Bull. London Math. Soc.

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The exponent of matrix multiplication is the smallest constant ω such that two n × n matrices may be multiplied by performing O(n ω+ ) arithmetic operations for every > 0. Determining the constant ω is a central question in both computer science and mathematics. Strassen [Linear Algebra Appl. 52/53 (1983) 645-685] showed that ω is also governed by the tensor rank of the matrix multiplication tensor. We define certain symmetric tensors, that is, cubic polynomials, and our main result is that their symmetric rank also grows with the same exponent ω, so that ω can be computed in the symmetric setting, where it may be easier to determine. In particular, we study the symmetrized matrix multiplication tensor sM n defined on an n × n matrix A by sM n (A) = trace(A 3 ). The use of polynomials enables the introduction of additional techniques from algebraic geometry in the study of the matrix multiplication exponent ω.

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“…The proof of the above theorem relies on the description of the relevant Hilbert scheme given in [23].…”

Section: Introductionmentioning

confidence: 99%

Distinguishing secant from cactus varieties

Gałązka¹,

Mańdziuk²,

Rupniewski³

2020

Preprint

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In this paper we study homogeneous polynomials and vector subspaces of a polynomial ring that are divisible by a large power of a linear form. We compute their cactus and border cactus ranks. We show that for d ≥ 5, the component of the cactus variety κ14(ν d (P 6 )) other than the secant variety σ14(ν d (P 6 )) consists of degree d polynomials divisible by (d − 3)-th power of a linear form. For d ≥ 6 we present an algorithm for deciding whether a point in the cactus variety κ14(ν d (P 6 )) belongs to the secant variety σ14(ν d (P 6 )). Analogously, we show that for d ≥ 5, the component of the Grassmann cactus variety κ8,3(ν d (P 4 )) other than the Grassmann secant variety σ8,3(ν d (P 4 )) consists of subspaces divisible by (d − 2)-th power of a linear form. Finally, for d ≥ 5 we present an algorithm for deciding whether a point in the Grassmann cactus variety κ8,3(ν d (P 4 )) belongs to the Grassmann secant variety σ8,3(ν d (P 4 )).

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VSPs of cubic fourfolds and the Gorenstein locus of the Hilbert scheme of 14 points on A6

Cited by 12 publications

References 29 publications

Identifying limits of ideals of points in the case of projective space

Identifying limits of ideals of points in the case of projective space

Polynomials and the exponent of matrix multiplication

Distinguishing secant from cactus varieties

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