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Determinantal variety

The radical ideal defining the determinantal variety is generated by the (r + 1) × (r + 1) minors of the matrix (Bruns-Vetter, Theorem 2.10).

Version
v1 · 2026-09-28 · History
Domain-specific #
8936
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Algebraic Geometry, Commutative Algebra → Mathematics

Core Idea

Determinantal variety is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: The radical ideal defining the determinantal variety is generated by the (r + 1) × (r + 1) minors of the matrix (Bruns-Vetter, Theorem 2.10). In algebraic geometry, determinantal varieties are spaces of matrices with a given upper bound on their ranks. Their significance comes from the fact that many examples in algebraic geometry are of this form, such as the Segre embedding of a product of two projective spaces.

How would you explain it like I'm…

The Copycat Grid Club

Think of all the grids of numbers of a certain size. Some grids are 'copycat' grids: their rows are really just made by mixing a few special rows together. The determinantal variety is the club of all grids that are copycat enough, using at most a chosen number of truly different rows. There is a test for joining: cut out every little square patch of a certain size, do a special calculation on it, and if every patch gives zero, the grid is in the club.

Low-Rank Matrix Collection

A matrix is a rectangle of numbers, and its rank counts how many truly different directions its rows point in, after you remove ones you could build from the others. A determinantal variety is the set of all matrices of a given size whose rank is at most some number r. The neat part is that you can test membership with equations: take every little square block of size r+1 inside the matrix, compute its determinant, and check that all of them are zero. Because it is described by equations like this, mathematicians can study it as a geometric shape.

Rank-Bounded Matrix Variety

Fix sizes m and n and a number r. The determinantal variety is the set of all m×n matrices over a field whose rank is at most r. A matrix has rank at most r exactly when every (r+1)×(r+1) minor (the determinant of an (r+1)×(r+1) square sub-block) equals zero, so the set is cut out by polynomial equations and counts as an algebraic variety. If you treat the matrix entries as independent variables, each of these minors is a polynomial of degree r+1. A key theorem says that these minors generate the radical ideal of the variety, meaning they capture all the polynomial equations it satisfies, not just some of them. Many classic examples in algebraic geometry, like the Segre embedding of a product of projective spaces, turn out to be of this form.

 

Given m, n and r, the determinantal variety is the locus of m×n matrices over a field k with rank ≤ r. Take the generic matrix whose entries x_ij are algebraically independent variables; its (r+1)×(r+1) minors are homogeneous polynomials of degree r+1 in k[x_ij], and the ideal they generate is called a determinantal ideal. Since a matrix has rank ≤ r precisely when all these minors vanish, the locus is an algebraic variety. The load-bearing result (Bruns–Vetter, Theorem 2.10) is that this determinantal ideal is radical, so it is exactly the ideal of all polynomials vanishing on the variety, not merely an ideal with the right zero set. Many standard constructions in algebraic geometry are determinantal in this sense, including the Segre embedding of a product of two projective spaces.

Scope of Application

  • Definition. Given m and n and r r is the set of all m × n matrices (over a field k) with rank ≤ r.

  • Definition. This is naturally an algebraic variety as the condition that a matrix have rank ≤ r is given by the vanishing of all of its (r + 1) × (r + 1) minors.

  • Definition. Considering the generic m × n matrix whose entries are algebraically independent variables x i,j , these minors are polynomials of degree r + 1.

  • Definition. The ideal of k[x i,j ] generated by these polynomials is a determinantal ideal.

  • Definition. Since the equations defining minors are homogeneous, one can consider Y r either as an affine variety in mn-dimensional affine space, or as a projective variety in (mn − 1)-dimensional projective.

Clarity

A clear use of Determinantal variety names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is The radical ideal defining the determinantal variety is generated by the (r + 1) × (r + 1) minors of the matrix (Bruns-Vetter, Theorem 2.10).

Manages Complexity

Determinantal variety compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—the ideal of k[x i,j ] generated by these polynomials is a determinantal ideal.—and the practical consequence—one can "globalize" the notion of determinantal varieties by considering the space of linear maps between two vector bundles on an algebraic variety.

Abstract Reasoning

  1. Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
  2. State the relation. Use the source-grounded identity: The radical ideal defining the determinantal variety is generated by the (r + 1) × (r + 1) minors of the matrix (Bruns-Vetter, Theorem 2.10).
  3. Check operation and conditions. The radical ideal defining the determinantal variety is generated by the (r + 1) × (r + 1) minors of the matrix (Bruns-Vetter, Theorem 2.10).
  4. Demand recognition evidence.

Knowledge Transfer

Within the home domain. Knowledge about Determinantal variety transfers literally when a new case preserves the same carrier type, relation, and recognition test. Given m and n and r r is the set of all m × n matrices (over a field k) with rank ≤ r. This is naturally an algebraic variety as the condition that a matrix have rank ≤ r is given by the vanishing of all of its (r + 1) × (r + 1) minors. Beyond the home domain. No canonical parent is asserted for Determinantal variety.

Relationships to Other Abstractions

Local relationship map for Determinantal varietyParents appear above the current abstraction, mutual partners to the right, and children below. Node labels state whether each abstraction is prime or domain-specific; colors identify relation types.Determinantal varietyDOMAINDomain-specific abstraction: Algebraic Variety — is a kind ofAlgebraicVarietyDOMAIN

Current abstraction Determinantal variety Domain-specific

Parents (1) — more general patterns this builds on

  • Determinantal variety is a kind of Algebraic Variety Domain-specific

    A determinantal variety is an algebraic variety cut out by minors imposing a matrix-rank bound.

Hierarchy path (1) — routes to 1 parentless root

Neighborhood in Abstraction Space

Determinantal variety sits in a moderately populated region (41st percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Polynomials & Algebraic Invariants (20 abstractions)

Nearest neighbors

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