Toward a salmon conjecture

TL;DR

This paper uses numerical algebraic geometry to identify specific polynomials that define certain secant varieties, advancing understanding of their algebraic structure and contributing to the Salmon conjecture in algebraic geometry.

Contribution

It demonstrates that particular degree 6 and 9 polynomials define the secant variety $\sigma_{4}(P^{2} imes P^{2} imes P^{3})$, extending to larger varieties relevant to the Salmon conjecture.

Findings

Identified degree 6 and 9 polynomials defining the secant variety $\sigma_{4}(P^{2} imes P^{2} imes P^{3})$

Extended the set-theoretic defining equations to larger secant varieties including $\sigma_{4}(P^{3} imes P^{3} imes P^{3})$

Provided computational evidence supporting the set-theoretic equations for these varieties

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.