Vector bundles give equations of cactus varieties

We present scheme theoretic methods that apply to the study of secant varieties. This mainly concerns finite schemes and their smoothability. The theory generalises to the base fields of any characteristic, and even to non-algebraically closed fields. In particular, the smoothability of finite schemes does not depend on the embedding into a smooth variety or on base field extensions. Independent of the base field, secant varieties to high degree Veronese reembeddings behave well with respect to the intersection and they are defined by minors of catalecticants whenever a suitable smoothability condition for Gorenstein subschemes holds. The content of the article is largely expository, although many results are presented in a stronger form than in the literature. addresses:J. Buczyński,

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“…5.1.1]. In fact, even larger set, called a cactus variety is contained in the same locus [Gał16,Thm 4] or Theorem 7.5.…”

Section: Secant Varietiesmentioning

confidence: 99%

Finite schemes and secant varieties over arbitrary characteristic

Buczyński¹,

Jelisiejew²

2017

Differential Geometry and its Applications

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“…See also [48,5]. As far as we are aware, however, the equations coming from minors of Young flattenings do not separate cactus rank from border rank [27].…”

Section: Introductionmentioning

confidence: 99%

Border Ranks of Monomials

Oeding

2016

Preprint

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Young flattenings, introduced by Landsberg and Ottaviani, give determinantal equations for secant varieties and their non-vanishing provides lower bounds for border ranks of tensors and in particular polynomials. We study monomial-optimal shapes for Young flattenings, which exhibit the limits of the Young flattening method. In particular, they provide the best possible lower bound for large classes of monomials including all monomials up to degree 6, monomials in 3 variables, and any power of the product of variables. On the other hand, for degree 7 and higher there are monomials for which no Young flattening can give a lower bound that matches the conjecturally tight upper bound of Landsberg and Teitler.

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Vector bundles give equations of cactus varieties

Cited by 2 publications

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Finite schemes and secant varieties over arbitrary characteristic

Finite schemes and secant varieties over arbitrary characteristic

Border Ranks of Monomials

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