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The set-theoretic version of the Salmon Conjecture (that is, finding the equations that cut out the fourth secant variety of the Segre embedding of $\mathbb P^3 \times \mathbb P^3 \times \mathbb P^3$ set-theoretically) has been resolved by S. Friedland and E. Gross in https://arxiv.org/abs/1104.1776

In view of this, I am wondering if resolving the scheme-theoretic version of the Salmon conjecture (that is, finding the defining ideal of the above variety) 1) is much harder/needing new ideas compared to the set-theoretic solution and 2) carries more information for the biological applications that motivated the conjecture than the set-theoretic version.

Also, is the problem of finding the ideals of other secant varieties of Segre embeddings 1) expected to be resolved in the near future and 2) relevant for biological or other applications? What are some approachable/interesting problems here?

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    $\begingroup$ The scheme-theoretic version is still open, so it seems that it may need a new idea. Note, that is distinct from finding the defining ideal (scheme-theoretic is the ideal up to a radical). It seems difficult to guess how much harder it will be, or whether the "near future" might hold resolutions for more general versions of the question. I have no idea if ideals or schemes matter for biological applications, but there are connections with geometric complexity theory, see e.g., arxiv.org/abs/1305.7387, or the book that Landsberg wrote called "Geometry and Complexity Theory". $\endgroup$ Jan 13, 2020 at 6:04
  • $\begingroup$ Thanks a lot! I am not sure that the question (or its meaning) about equations defining secant varieties of Segre varieties appears in the above Landsberg's writings explicitly. I understand that we want to study the rank or border rank of the matrix multiplication tensor, so we want to understand which secant variety of the Segre variety of $\mathbb P^n \times \mathbb P^n \times \mathbb P^n$ it lies in, is this why we want to study the equations of these secant varieties? Even if we knew the equations, probably it would not be easy to see if the matrix multiplication tensor satisfies them? $\endgroup$
    – Yellow Pig
    Jan 13, 2020 at 9:36
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    $\begingroup$ Oops, scheme-theoretic means finding the defining ideal up to saturation, not up to radical. Sorry. ... Besides that: Landsberg has definitely written about equations of secant varieties (and so have many others) but maybe not in his GCT writing. And I think you're right, only finding equations isn't necessarily enough - they have to also be ones that you can evaluate on the matrix multiplication tensor, and that's a nontrivial problem. $\endgroup$ Jan 13, 2020 at 17:22

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