Are all Secant Varieties of Segre Products Arithmetically Cohen-Macaulay?

Abstract

When present, the Cohen-Macaulay property can be useful for finding the minimal defining equations of an algebraic variety. It is conjectured that all secant varieties of Segre products of projective spaces are arithmetically Cohen-Macaulay. A summary of the known cases where the conjecture is true is given. An inductive procedure based on the work of Landsberg and Weyman (LW-lifting) is described and used to obtain resolutions of orbits of secant varieties from those of smaller secant varieties. A new computation of the minimal free resolution of the variety of border rank 4 tensors of format $3 \times 3 \times 4$ is given together with its equivariant presentation. LW-lifting is used to prove several cases where secant varieties are arithmetically Cohen-Macaulay and arithmetically Gorenstein.