Toward a salmon conjecture

Details Toward a salmon conjecture Toward a salmon conjecture

2010-09-30

arxiv.org/abs/1009.6181v2

Mathematics

Algebraic Geometry

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

Scientific paper

10.1080/10586458.2011.576539

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.

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