A tensor's rank decomposition expresses it as a sum of rank-one tensors. The decomposition is identifiable if it's unique (up to permutation and scaling). Identifiability is crucial for applications — signal processing, psychometrics, chemometrics — because a non-identifiable decomposition means the factors can't be uniquely recovered from the data.
The strong base locus framework connects identifiability to projective geometry. The base locus of a linear system on a Veronese or Segre-Veronese variety determines where tangential projections fail to be birational. At points in the base locus, the tangential projection drops rank, meaning the tensor at that point has a non-unique decomposition.
The new framework introduces base loci for tangential projections specifically, relating them to interpolation problems on the variety. A tensor is identifiable if and only if it lies outside the base locus of the relevant tangential projection — a geometric condition that can be checked using the intersection theory of the variety.
For Veronese varieties (symmetric tensors), the base locus is controlled by the degree and the number of summands. For Segre-Veronese varieties (partially symmetric tensors), the base locus depends on the multi-degree and the partition structure.
The practical payoff: identifiability tests reduce to checking membership in a geometric locus. The geometry of the variety — its intersection theory, its tangent spaces — determines which tensors have unique decompositions. The algebra of tensors is controlled by the geometry of the varieties that parameterize them.