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  "body": "\n## Definition\n\nLet $k$ be a field, let $1 \\le d \u003c n$, and let $\\mathrm{Gr}(d, n)$ be the [[Grassmannian|CYPA882RB9]] of $d$-dimensional subspaces of $k^n$.\nFix a full flag $F_1 \\subset F_2 \\subset \\cdots \\subset F_n = k^n$ with $\\dim F_j = j$.\nLet $\\lambda = (\\lambda_1, \\ldots, \\lambda_d)$ be a partition with $n - d \\ge \\lambda_1 \\ge \\cdots \\ge \\lambda_d \\ge 0$.\nThe *Schubert variety* $X_\\lambda$ is\n$$X_\\lambda = \\set{\\Lambda \\in \\mathrm{Gr}(d, n) : \\dim(\\Lambda \\cap F_{n - d + i - \\lambda_i}) \\ge i \\text{ for } i = 1, \\ldots, d}.$$\n\n## Remarks\n\n$X_\\lambda$ is a closed irreducible subvariety of codimension $\\abs{\\lambda} = \\lambda_1 + \\cdots + \\lambda_d$.\n\nExample: $\\lambda = 0$ gives all of $\\mathrm{Gr}(d, n)$.\nIndeed $\\dim(\\Lambda \\cap F_{n - d + i}) \\ge d + (n - d + i) - n = i$ holds for every $\\Lambda$.\nFor $d = 2$, $n = 4$ and $\\lambda = (1, 0)$, the conditions reduce to $\\dim(\\Lambda \\cap F_2) \\ge 1$.\nIn projective terms these are the lines in $\\mathbb{P}^3$ that meet a fixed line, a hypersurface in $\\mathrm{Gr}(2, 4)$.\n\nNon-example: the set of $\\Lambda \\in \\mathrm{Gr}(2, 4)$ with $\\Lambda \\cap F_2 = 0$ is open and dense, and it is not closed.\nSo it is not a Schubert variety; it is the open Schubert cell.\n\nGeneral form: let $G$ be a connected reductive group with a [[Borel subgroup|5JSYA1JNCC]] $B$ and a parabolic subgroup $P \\supseteq B$.\nA Schubert variety in the [[flag variety|NHRWCQQKTH]] $G / P$ is the Zariski closure of a $B$-orbit.\n",
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