On the Ideals of Secant Varieties of Segre Varieties

Abstract

We establish basic techniques for determining the ideals of secant varieties of Segre varieties. We solve a conjecture of Garcia, Stillman, and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture set-theoretically for an arbitrary number of factors. We determine the low degree components of the ideals of secant varieties of small dimension in a few cases.

Lectures on Algebraic Statistics

“…This was conjectured in [50, §7] and proved in [5] and in [65]. The DraismaKuttler theorem [36] can be regarded as a generalization of this statement.…”

“…However, for the most interesting case of DNA sequences (r = 4), it is still an open problem to describe the generating sets of these ideals. It is known that certain polynomials of degrees 5, 6, and 9 are needed as generators, but it is unknown whether these polynomials suffice [65,88].…”

On the ideals of equivariant tree models

“…The main missing ingredients for successful applications are equations for star models. These are very hard to come by: [9] posed several conjectures concerning these for the general Markov model, and special cases of these conjectures were proved in [1,13,14]. For certain important equivariant models equations were found in [4,16].…”

Geometry and the complexity of matrix multiplication

“…First, in order to prove that a set of equations E is a set of defining equations for σ r (Seg(PA 1 × · · · × PA n )), one must prove that any point in the zero set of E is either a point on a secant P r−1 or on a limit P r−1 . For example, the proof of the set-theoretic GSS conjecture (see §12) in [37] proceeded in this fashion. Second, to establish lower bounds for the border rank of a given tensor, e.g., matrix multiplication, one could try to first prove that it cannot lie on any secant P r−1 and then that it cannot lie on any limiting P r−1 either.…”

“…Once one has an explicit description of a space of polynomials as modules, it is algorithmic to write down an explicit basis of the module as we did in §10.1. See [37,40] for more details. Given any G-module W , the first thing to do when studying W is to try to decompose it into isotypic components (which is always possible when G is reductive, as is our situation).…”