When is the set of tensors of rank at most $r$ closed?

Question

Consider the vector space $V=A_1\otimes A_2\otimes\dots\otimes A_k$ where $k\geq 2$ and let $\sigma_r$ be the set of tensors in $V$ of rank at most $r$. It seems clear that $\sigma_r$ is Zariski (Euclidean?) closed in the following cases :

1 - $k=2$ and $r$ is anything

2 - $r=1$ and $k$ is anything

3 - $\sigma_r=V$

Is this list complete? Can I change the third case as $r=\prod_{i=1}^k \dim A_i$ and if not with some other integral condition on $r$?

Answer 1

1. First of all, the maximal possible rank of degree $k$ tensors is bounded from above by $$ R=\prod_{i=1}^{k-1} dim(A_i). $$ Here and below the indices are arranged so that $$2\le n_1= dim(A_1)\le n_2= dim(A_2)\le ...\le n_k= dim(A_k).$$

See for instance

M.V. Catalisano, A.V. Geramita, A. Gimigliano, "Ranks of tensors, secant varieties of Segre varieties and fat points", Linear Algebra Appl. 355 (2002) 263-285.

This bound is better than the one that you wrote in the last part of your question. In particular, $\sigma_R$ equals the entire tensor product $$ V= A_1\otimes ... \otimes A_k, $$ which is, of course, Zariski closed.

2. There are some Zariski closed examples besides the ones that you listed. For instance (I am working over ${\mathbb C}$), consider triples $$ (n_1,n_2,n_3)= (2, 2, n), n\ge 4. $$ Then in $V={\mathbb C}^2 \otimes {\mathbb C}^2 \otimes {\mathbb C}^n$, all tensors of rank =4 also have the border rank =4, see Table 10.3.1 on page 253 of

J.Landsberg, "Tensors: Geometry and Applications", AMS, 2011.

This means that the set $\sigma_3\subset V$ of tensors of rank $\le 3$ is closed. (In contrast, $\sigma_2$ is not closed.) At the same time, since $n\ge 4$, $V=\sigma_4\ne \sigma_3$.

I very much doubt that a complete list of tuples $(n_1,...,n_k,r)$ for which $\sigma_r$ is Zariski closed is known. Among general results on lack of closedness in the classical topology the following is known, see

V. De Silva, L.-H. Lim, "Tensor rank and the ill-posedness of the best low-rank approximation problem," SIAM J. Matrix Anal. Appl. 30 (2008) 1084-1127.

Theorem. Suppose $k\ge 3$ and $$ 2\le r\le \min(n_1,...,n_k). $$ Then $\sigma_r$ is not closed in the classical topology. (Hence, it is not closed in the Zariski topology either.)