5.8.4 Tensor Decomposable Tensor Rank Relation

Tensor Decomposable Tensor Rank Relation is the geometric restatement, in the language of secant varieties, of the elementary fact that decomposable tensors are exactly the rank-one tensors: the decomposable locus is the base variety whose successive secant varieties define the loci of tensors of rank at most two, three, and so on, making decomposability the seed from which the entire rank stratification of a tensor product space is built.

The Base Case

For a tensor product T = V1 ⊗ V2 ⊗ ⋯ ⊗ Vn over a field F, an element t is decomposable exactly when its tensor rank satisfies

This is immediate from the definitions: rank(t) is the fewest number of decomposable summands needed to write t as a sum, so rank at most one means a single decomposable term already equals t, which is precisely decomposability.

Secant Varieties Built on the Decomposable Locus

The rank relation extends beyond the base case through the geometric device of secant varieties, which describe higher-rank loci directly in terms of the decomposable locus.

Definition of the r-th Secant Variety

Writing X for the decomposable locus (the projectivization of all decomposable tensors) inside the projective space of T, the r-th secant variety σr(X) is the Zariski closure of the union of all linear spans of r points chosen on X. Concretely, a generic element of σr(X) is a sum of r decomposable tensors, matching the definition of a tensor of rank exactly r.

Rank Loci as Secant Varieties

The locus of tensors of rank at most r coincides, up to the technical distinction between rank and border rank, with σr(X). In particular, σ1(X) = X recovers the decomposable locus itself as the first and smallest term in this nested chain of varieties,

reaching the whole ambient projective space once r is large enough.

Border Rank and the Closure Subtlety

The decomposable-tensor rank relation becomes more delicate once the closure operation implicit in the definition of a secant variety is taken seriously.

Border Rank

An element t has border rank at most r if it lies in the secant variety σr(X) as defined by Zariski closure, even if t itself is not literally a sum of r decomposable tensors — only a limit of such sums. Border rank is always at most ordinary tensor rank, and for tensors of three or more factors, the two invariants can genuinely differ.

Non-Closedness of Higher Rank Loci

For n ≥ 3, the set of tensors of rank at most r can fail to be Zariski closed, meaning some tensors of rank strictly greater than r are limits of sequences of tensors of rank exactly r. This is a well-known example of "border rank jumping," and it means the naive rank-based description of the decomposable locus's secant varieties requires the more careful, closure-respecting border rank in order to remain geometrically well-behaved.

Dimension Counts and Generic Rank

The decomposable tensor rank relation also governs the expected (generic) rank of a tensor through the dimensions of the successive secant varieties.

Expected Dimension

The expected dimension of σr(X) is r times the dimension of X plus (r − 1), capped at the dimension of the full ambient space; when this expected dimension first meets or exceeds the ambient dimension, the corresponding r is expected to be the generic rank of a tensor in T.

Defective Secant Varieties

In certain special cases, the actual dimension of σr(X) falls short of this expected value — a phenomenon called defectivity — and when this happens, the generic tensor rank exceeds the naive dimension-count prediction. Classifying which decomposable loci give rise to defective secant varieties is an active area connecting tensor rank theory directly to classical projective algebraic geometry.

Illustrative Diagram

The solid curve is the decomposable locus X itself, and the dashed envelope around it represents the second secant variety, built from spans of pairs of points on X, illustrating how higher-rank loci are generated directly from the decomposable base.