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.
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.
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.
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.
Relationships to Other Abstractions¶
Current abstraction Secant Variety Domain-specific
Parents (1) — more general patterns this builds on
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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 (39th 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
- Algebraic Surface — 0.91
- Degree of an algebraic variety — 0.90
- Motive (algebraic geometry) — 0.87
- Rational normal curve — 0.87
- Rational Normal Scroll — 0.87
Computed from structural-signature embeddings · 2026-10-08