15.20.3 Tensor Symmetric Tensor Geometry Role

Tensor Symmetric Tensor Geometry Role is the identification of symmetric tensors, considered up to scalar multiple, with points of a projective variety, the Veronese variety, and the resulting translation of decomposition questions about symmetric tensors into questions about the linear-algebraic and incidence geometry of secant varieties built from this projective variety.

Symmetric Tensors as Points of Projective Space

Projectivizing the Symmetric Power

The space of symmetric tensors of order d on an n-dimensional vector space, S^d V, is itself a vector space, and considering its nonzero elements up to scalar multiple produces a projective space, whose dimension is one less than the dimension of S^d V computed via the binomial coefficient discussed under the Symmetric Power Notation. Every symmetric tensor, aside from the origin, thus determines a well-defined point of this projective space, and the Geometry Role is the systematic study of symmetric tensors through the properties of the points, curves, and subvarieties they determine within it.

The Veronese Variety as the Locus of Pure Powers

Within this projective space, the points corresponding to pure power forms, the rank-one symmetric tensors discussed under Tensor Symmetric Decomposition Pure Power Form, trace out a distinguished subvariety known as the Veronese variety, obtained as the image of the projective space of the underlying vector space V under the map sending a vector's projective class to the projective class of its d-th tensor power. The Veronese variety is smooth, irreducible, and has dimension equal to that of the underlying projective space of V, sitting inside the much larger ambient projective space of all symmetric tensors.

Secant Varieties and Decomposition

Secant Varieties as the Geometric Encoding of Rank

The k-th secant variety of the Veronese variety is defined as the closure of the union of all linear spans of k points chosen on the Veronese variety, and a symmetric tensor's projective class lies on this secant variety precisely when the tensor can be written, or approximated by tensors written, as a sum of k pure power forms. This geometric reformulation is exact: the symmetric rank of a tensor, as defined in the Tensor Symmetric Decomposition Rank Relation, is the smallest k for which the tensor's projective class lies on the k-th secant variety, and the symmetric border rank is the smallest k for which it lies on the Zariski closure of that same union.

Dimension Counting and Expected Rank

Because the expected dimension of the k-th secant variety can be computed by a naive parameter count, comparing the dimension of a generic union of k tangent or linear spaces against the dimension of the ambient projective space, the Geometry Role supplies a direct route to predicting the generic symmetric rank of tensors of a given order and dimension, before any decomposition is attempted; the Alexander-Hirschowitz theorem is precisely the classification of the finitely many exceptional cases in which the actual dimension of a secant variety falls short of this naive expectation, a phenomenon called secant defectivity.

The Term Set as an Incidence Configuration

Points, Spans, and Decomposition

A minimal symmetric decomposition of a tensor T, as described under the Term Set concept, corresponds geometrically to a set of points on the Veronese variety, of minimal cardinality, whose linear span contains the projective point corresponding to T; this reframes the Reconstruction problem as a question of incidence geometry, seeking a specific configuration of points on a known variety passing through, or spanning a space containing, a given target point.

Apolarity as a Dual Geometric Description

The apolar ideal used throughout Reconstruction and rank computation has a direct geometric meaning: it is the ideal of forms vanishing on the sought configuration of points, and the geometric problem of finding this configuration, known as finding an apolar set, is dual, via the pairing between a vector space and its dual, to the direct search for points on the Veronese variety spanning the target tensor, giving two equivalent geometric routes to the same decomposition.

Consequences of the Geometric Perspective

Uniqueness as an Identifiability Statement About Secant Varieties

The uniqueness or non-uniqueness of a minimal Term Set, discussed as an algebraic question under that heading, is understood geometrically as a statement about the fibers of the map that sends a configuration of k points on the Veronese variety to its linear span: identifiability corresponds to this map being generically injective (a finite fiber consisting of a single point) at the relevant rank, while non-identifiability corresponds to the fiber being positive-dimensional, and results on secant variety non-defectivity and identifiability, refining the Alexander-Hirschowitz classification, are the geometric counterpart of the uniqueness theory developed for the Term Set.

Bridge to Rank Relation and Border Rank Comparisons

The comparison between symmetric rank and ordinary tensor rank, addressed by the Rank Relation, is recast geometrically as a comparison between the secant varieties of the Veronese variety and the secant varieties of the Segre variety, the analogous projective variety parametrizing ordinary rank-one tensors, and the known counterexamples to Comon's Conjecture are, in this geometric language, instances where a point lies on a low secant variety of the Segre variety while requiring a strictly higher secant variety of the Veronese variety to be reached, making the Geometry Role the natural setting in which the entire decomposition theory of symmetric tensors is most transparently unified.