  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.
