Border rank of ternary trilinear forms and the j-invariant

Allums, Derek; Landsberg, Joseph M.

doi:10.2140/involve.2015.8.345

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

Derek Allums and Joseph M. Landsberg

Vol. 8 (2015), No. 2, 345–355

DOI: 10.2140/involve.2015.8.345

Abstract

We first describe how one associates a cubic curve to a given ternary trilinear form $ϕ \in ℂ^{3} \otimes ℂ^{3} \otimes ℂ^{3}$ . We explore relations between the rank and border rank of the tensor $ϕ$ 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.

Keywords

algebraic geometry, border rank of tensors, $j$-invariant of cubic, ternary trilinear forms

Mathematical Subject Classification 2010

Primary: 15A72, 68Q17

Milestones

Received: 18 September 2013

Accepted: 24 January 2014

Published: 3 March 2015

Communicated by David Royal Larson

Authors

Derek Allums
Department of Mathematics Rice University Houston, TX 77005 United States

Joseph M. Landsberg
Texas A&M University College Station, TX 77843 United States

Vol. 8, No. 2, 2015