Secant Variety¶

The Zariski closure of the union of linear spans of k+1 points of an embedded projective variety; for k=1 it closes all secant lines and their tangent limits.

Version: v1 · 2026-09-28 · History
Domain-specific #: 11928
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Projective Algebraic Geometry, Algebraic Geometry → Mathematics
Aliases: Variety of secants, Secant locus, Higher secant variety

Core Idea¶

For a projective variety V inside P^r, the first secant construction draws the projective line through each pair of points and takes the union. That union is then closed in the Zariski topology, producing an algebraic variety and adding limiting configurations such as tangent lines when the pair coalesces.

The kth secant variety repeats the construction with the projective span of k+1 points. Parameter counting gives expected dimension at most min(r,(k+1)dim V+k), but special geometry can lower actual dimension; such cases are secant defective. Terracini's lemma computes the tangent space at a general secant point from tangent spaces at the chosen points and is central to diagnosing the dimension.

Structural Signature¶

Sig role-phrases:

Projective variety V — Supplies the geometric locus whose point configurations are spanned. It is required carrier. Counterfactual: A point cloud without algebraic closure data is not the same object.
Ambient projective space — Provides lines, spans, dimension ceiling, and Zariski topology. It is required context. Counterfactual: Changing embedding can change the secant variety.
k+1 point configuration — Selects the points whose projective span contributes. It is required generator. Counterfactual: A kth secant span cannot be formed from an unspecified rank.
Linear span — Adds every point on the projective subspace through the configuration. It is defining operation. Counterfactual: Taking only chords' endpoints would omit the secant locus.
Zariski closure — Adds limiting configurations including tangential degenerations and makes an algebraic variety. It is required closure. Counterfactual: The raw union need not be closed.
Expected versus actual dimension — Diagnoses filling, defect, and singular structure. It is characteristic invariant. Counterfactual: Point-count intuition alone can overestimate dimension because fibers and dependencies occur.

What It Is Not¶

A secant variety is not merely the set of original points or pair endpoints.
It is not the raw union of spans when that union is not Zariski closed.
It is not only the tangent variety, although tangent directions appear as limits.
A numerical low-rank approximation set is not automatically the algebraic secant variety until embedding and closure are specified.
Closest near-miss. The tangent variety uses tangent spaces at single points; it lies in or relates to secant limits but is not generally the whole secant variety.

Scope of Application¶

Projective geometry. Secant dimension, singularity, and defect expose geometry of an embedded variety.
Tensor rank. Segre and related embeddings connect secant membership with border rank.
Identifiability. Fibers of span parameterizations help determine whether decompositions are unique.
Projection and embedding. Avoiding secant loci can ensure a projection remains injective on a variety.

Clarity¶

The notation convention for kth secant must state whether k counts points or secant order; here k+1 points span the kth variety. Exact rank corresponds to the constructible union, while border rank corresponds to its Zariski closure. The embedding and coefficient field matter, and filling the ambient space is an outcome rather than failure of definition.

Manages Complexity¶

The secant variety compresses all point configurations and linear combinations into one closed geometric locus. This turns decomposition and approximation questions into dimension, tangent-space, singularity, and equation problems. Closure improves algebraic tractability while deliberately adding limiting points that may lack an exact decomposition of the nominal size.

Abstract Reasoning¶

Fix the embedded projective variety, ambient space, field, and secant-index convention.
Choose k+1 general points and form their projective span.
Vary configurations and take the union of resulting spans.
Take Zariski closure and separate exact-rank from border-rank membership.
Compare actual dimension with the expected parameter bound.
Use Terracini's lemma or equations to analyze defect, singularity, and identifiability.

Knowledge Transfer¶

The construction transfers among projective embeddings, including varieties representing matrices, tensors, or forms. Ordinary Euclidean chords provide intuition but do not supply Zariski closure or projective rank meaning. Any application must identify what points and spans represent and whether exact or limiting decomposition is being asked.

Examples¶

Canonical¶

For a projective curve, chords through pairs of points sweep a locus whose Zariski closure also contains tangent-line limits at coincident points.

Mapped back: carrier → projective curve; closure → tangent limits; rank → two points; span → line.

Applied / In Practice¶

For k, tangent spaces at k+1 general points are combined via Terracini's lemma to calculate actual dimension and detect defect.

Mapped back: comparison → actual versus expected dimension; configuration → general points; tool → tangent-space span.

Structural Tensions¶

T1 — Constructible Union versus Algebraic Closure. Actual spans describe generic rank while closure adds limiting border-rank points.

Diagnostic: Is the question about exact decompositions or membership in the closed variety?

T2 — Expected Dimension versus Geometric Defect. Parameter counting predicts a bound, but special geometry can make the image smaller.

Diagnostic: What tangent-space dependence explains the dimension drop?

Structural–Framed Character¶

Secant Variety is strongly structural. Projective span, Zariski closure, dimension, and tangent spaces determine the object. The chosen embedding and field frame which secant variety is obtained, but evaluation is mathematical rather than social.

Structural Core vs. Domain Accent¶

The skeleton is closing the union of low-cardinality spans. Algebraic geometry supplies projective varieties, Zariski topology, dimension, tangent spaces, singularities, and equations. Removing those yields generic interpolation or convex-span reasoning.

This entry is a kind of Projective variety.

Approved root. No reviewed parent entails this projective span-and-closure construction.
Related — closure, span, rank, and approximation. They illuminate uses without asserted parent edges.

Relationships to Other Abstractions¶

Current abstraction Secant Variety Domain-specific

Parents (1) — more general patterns this builds on

Secant Variety is a kind of Projective variety Domain-specific

A Secant Variety is the Projective Variety obtained by Zariski-closing spans of finite point sets from an embedded projective variety.

Hierarchy path (1) — routes to 1 parentless root

Secant Variety → Projective variety → Algebraic Variety

Neighborhood in Abstraction Space¶

Secant Variety sits in a crowded region of the domain-specific corpus (39^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Algebraic Varieties & Topological Invariants (27 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Tangent variety. Tell: Sweeps tangent spaces from one point and is generally only part of the secant limit structure.
Join of varieties. Tell: Generalizes spans using points from possibly different varieties; a secant is a repeated self-join.
Convex hull. Tell: Uses real convex combinations and Euclidean closure, not projective spans and Zariski closure.
Rank locus. Tell: Exact rank can be nonclosed; the secant variety more naturally captures bounded border rank.

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Secant_variety (revision 1353813601).

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.