Border rank of ternary trilinear forms and the $j$-invariant

Abstract

We first describe how one associates a cubic curve to a given ternary trilinear form $\varphi \in {ℂ}^3\otimes {ℂ}^3\otimes {ℂ}^3$. We explore relations between the rank and border rank of the tensor $\varphi$ and the geometry of the corresponding cubic curve. When the curve is smooth, we show there is no relation. When the curve is singular, normal forms are available, and we review the explicit correspondence between the normal forms, rank and border rank.