Vol. 8, No. 2, 2015

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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

We first describe how one associates a cubic curve to a given ternary trilinear form ϕ 3 3 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.

algebraic geometry, border rank of tensors, $j$-invariant of cubic, ternary trilinear forms
Mathematical Subject Classification 2010
Primary: 15A72, 68Q17
Received: 18 September 2013
Accepted: 24 January 2014
Published: 3 March 2015

Communicated by David Royal Larson
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