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  "body": "\n## Definition\n\nLet $k$ be a field, and let $m, n \\ge 1$ and $0 \\le r \u003c \\min(m, n)$ be integers.\nIdentify the space of $m \\times n$ matrices over $k$ with the affine space $\\mathbb{A}^{mn}$, with coordinates $x_{ij}$.\nThe *determinantal variety* $M_r$ is the set of $m \\times n$ matrices of [[rank|0ZC9YTXZRY]] at most $r$.\nIt 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 $(x_{ij})$.\n\n## Remarks\n\nA matrix has rank at most $r$ if and only if all its $(r + 1) \\times (r + 1)$ minors vanish, so the two descriptions agree.\nFor $k$ algebraically closed, $M_r$ is irreducible of dimension $r(m + n - r)$, and the ideal $I_{r+1}$ is prime.\nSince $M_r$ is defined by homogeneous equations, it also defines a projective variety in $\\mathbb{P}^{mn - 1}$.\n\nExample: $m = n = 2$ and $r = 1$.\n$M_1$ is the quadric cone $x_{11} x_{22} - x_{12} x_{21} = 0$ in $\\mathbb{A}^4$, of dimension $3 = 1 \\cdot (2 + 2 - 1)$.\n\nNon-example: the set of $2 \\times 2$ matrices of rank exactly $1$ is not a determinantal variety.\nIt is not closed, since its closure contains the zero matrix, which has rank $0$.\n",
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