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  "body": "\n## Summary\n\nA determinantal variety is the set of matrices of a fixed size whose [[rank|0ZC9YTXZRY]] is at most a given number.\nIt is cut out by the vanishing of all [[minors|0BT9BYWVRY]] of one size larger, which generate a [[determinantal ideal|SXD7X0WQH1]].\nThese varieties are irreducible, normal and Cohen–Macaulay, and they are basic examples of singular varieties.\nMany varieties in geometry, such as secant varieties and rational normal curves, are described by determinantal equations.\n\n## Explanations\n\n### L1 human\n\nA $2 \\times 2$ matrix $\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix}$ has rank at most $1$ exactly when $ad - bc = 0$.\nThis one equation defines a three-dimensional cone in the four-dimensional space of matrices, with its tip at the zero matrix.\nFor larger matrices, \"rank at most $r$\" is expressed by the vanishing of all $(r + 1) \\times (r + 1)$ minors.\nThe resulting sets are the determinantal varieties, and their only singular points are the matrices of rank smaller than $r$.\n",
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