---
tag: 4JEBSN709Q
kind: definition
name: Determinantal variety
msc: [14M12]
defines: 6WH08A4BTK
uses: [0ZC9YTXZRY, 0BT9BYWVRY, SXD7X0WQH1, VHFMQQYSTN]
sources:
  - url: https://en.wikipedia.org/wiki/Determinantal_variety
---

## Definition

Let $k$ be a field, and let $m, n \ge 1$ and $0 \le r < \min(m, n)$ be integers.
Identify the space of $m \times n$ matrices over $k$ with the affine space $\mathbb{A}^{mn}$, with coordinates $x_{ij}$.
The *determinantal variety* $M_r$ is the set of $m \times n$ matrices of [[rank|0ZC9YTXZRY]] at most $r$.
It is the [[algebraic set|VHFMQQYSTN]] defined by the [[determinantal ideal|SXD7X0WQH1]] $I_{r+1}$, generated by all $(r + 1) \times (r + 1)$ [[minors|0BT9BYWVRY]] of the matrix $(x_{ij})$.

## Remarks

A matrix has rank at most $r$ if and only if all its $(r + 1) \times (r + 1)$ minors vanish, so the two descriptions agree.
For $k$ algebraically closed, $M_r$ is irreducible of dimension $r(m + n - r)$, and the ideal $I_{r+1}$ is prime.
Since $M_r$ is defined by homogeneous equations, it also defines a projective variety in $\mathbb{P}^{mn - 1}$.

Example: $m = n = 2$ and $r = 1$.
$M_1$ is the quadric cone $x_{11} x_{22} - x_{12} x_{21} = 0$ in $\mathbb{A}^4$, of dimension $3 = 1 \cdot (2 + 2 - 1)$.

Non-example: the set of $2 \times 2$ matrices of rank exactly $1$ is not a determinantal variety.
It is not closed, since its closure contains the zero matrix, which has rank $0$.