Fano schemes for generic sums of products of linear forms

@article{Ilten2016FanoSF,
  title={Fano schemes for generic sums of products of linear forms},
  author={Nathan Owen Ilten and Hendrik Suss},
  journal={Journal of Algebra and Its Applications},
  year={2016}
}
We study the Fano scheme of [Formula: see text]-planes contained in the hypersurface cut out by a generic sum of products of linear forms. In particular, we show that under certain hypotheses, linear subspaces of sufficiently high dimension must be contained in a coordinate hyperplane. We use our results on these Fano schemes to obtain a lower bound for the product rank of a linear form. This provides a new lower bound for the product ranks of the [Formula: see text] Pfaffian and [Formula: see… 

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References

SHOWING 1-10 OF 13 REFERENCES

Lower bound for ranks of invariant forms

Apolarity for determinants and permanents of generic matrices

We show that the apolar ideals to the determinant and permanent of a generic matrix, the Pfaffian of a generic skew symmetric matrix and the Hafnian of a generic symmetric matrix are each generated

Equations for secant varieties of Chow varieties

  • Yonghui Guan
  • Mathematics, Computer Science
    Int. J. Algebra Comput.
  • 2017
The method of prolongation is used to obtain equations for secant varieties of Chow varieties as $ GL(V)$- modules as $GL(V),$-modules to prove Valiant's conjecture.

Lines on projective hypersurfaces

Abstract We study the Hilbert scheme of lines on hypersurfaces in the projective space. The main result is that for a smooth Fano hypersurface of degree at most 6 over an algebraically closed field

Flattenings and Koszul Young flattenings arising in complexity theory

ABSTRACT I find new equations for Chow varieties, their secant varieties, and an additional variety that arises in the study of complexity theory by flattenings and Koszul Young flattenings. This

The permanent of a square matrix

Foundations of the theory of Fano schemes

© Foundation Compositio Mathematica, 1977, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions

Fano schemes of determinants and permanents

Let $D_{m,n}^r$ and $P_{m,n}^r$ denote the subschemes of $\mathbb{P}^{mn-1}$ given by the $r\times r$ determinants (respectively the $r\times r$ permanents) of an $m\times n$ matrix of

Product Ranks of the 3 × 3 Determinant and Permanent

Abstract We show that the product rank of the $3\,\times \,3$ determinant ${{\det }_{3}}$ is $5$ , and the product rank of the $3\,\times \,3$ permanent $\text{per}{{\text{m}}_{3}}$ is $4$ . As a

On the Nuclear Norm and the Singular Value Decomposition of Tensors

  • H. Derksen
  • Mathematics, Computer Science
    Foundations of Computational Mathematics
  • 2015
Inspired by the heuristics of convex relaxation, the nuclear norm is considered instead of the rank of a tensor and thenuclear norm of various tensors of interest is determined.