Toward a salmon conjecture

Abstract

By using a result from the numerical algebraic geometry package Bertini we show that (up to high numerical accuracy) a specific set of degree 6 and degree 9 polynomials cut out the secant variety ${\sigma }_4({\mathbb{P}}^2\times {\mathbb{P}}^2\times {\mathbb{P}}^3)$. This, combined with an argument provided by Landsberg and Manivel (whose proof was corrected by Friedland), implies set-theoretic defining equations in degrees 5, 6 and 9 for a much larger set of secant varieties, including ${\sigma }_4({\mathbb{P}}^3\times {\mathbb{P}}^3\times {\mathbb{P}}^3)$ which is of particular interest in light of the salmon prize offered by E. Allman for the ideal-theoretic defining equations.

arXiv comment15 pages. Updated to reflect the referees' suggestions. Also added ancillary files to the arXiv in this version

References in corpus (4)

• Set Theoretic Defining Equations of the Variety of Principal Minors of Symmetric Matrices
• Symplectic bundles on the plane, secant varieties and Lüroth quartics revisited
• Set-theoretic defining equations of the tangential variety of the Segre variety
• Equations for secant varieties to Veronese varieties

Cited by in corpus (14)

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• Computations and Equations for Segre-Grassmann hypersurfaces
• Homotopy techniques for tensor decomposition and perfect identifiability
• Are all Secant Varieties of Segre Products Arithmetically Cohen-Macaulay?
• Noetherian operators and primary decomposition
• On tensors of border rank $l$ in $\C^{m\times n\times l}$
• Equations for lower bounds on border rank
• A proof of the set-theoretic version of the salmon conjecture
• Arithmetic properties of the first secant variety to a projective variety