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.

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

References in corpus (2)

• Symplectic bundles on the plane, secant varieties and Lüroth quartics revisited
• Equations for secant varieties to Veronese varieties

Cited by in corpus (14)

• Bounded-rank tensors are defined in bounded degree
• Algebraic Geometry tools for the study of entanglement: an application to spin squeezed states
• Four lectures on secant varieties
• Equations for the fifth secant variety of Segre products of projective spaces
• The Quadrifocal Variety
• Border Ranks of Monomials
• 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