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  "body": "\n## Definition\n\nLet $X$ be a smooth projective [[algebraic curve|XJM2MT4S3Y]] of [[genus|6W8TDV9VVN]] $g$ over an algebraically closed field.\nFor an integer $d$, let $\\operatorname{Pic}^d(X)$ be the set of isomorphism classes of [[line bundles|7JCF2TE3V5]] of degree $d$ on $X$.\nIt is a subset of the [[Picard group|3Z9DCWYXQZ]] of $X$.\nFor integers $r \\ge 0$ and $d$:\n\n1. The *Brill–Noether locus* is $W^r_d(X) = \\set{L \\in \\operatorname{Pic}^d(X) : h^0(X, L) \\ge r + 1}$, the set of line bundles of degree $d$ whose complete [[linear system|GHRPE8CXJ9]] has dimension at least $r$.\n2. The *Brill–Noether number* is $\\rho(g, r, d) = g - (r + 1)(g - d + r)$.\n\n*Brill–Noether theory* is the study of the loci $W^r_d(X)$, in particular of when they are nonempty and of their dimension.\n\n## Remarks\n\nThe locus $W^r_d(X)$ is a closed subvariety of $\\operatorname{Pic}^d(X)$, which is a copy of the Jacobian, of dimension $g$.\nThe number $\\rho$ is the expected dimension of $W^r_d(X)$: it is $g$ minus the number of conditions $(r + 1)(g - d + r)$ that the existence of $r + 1$ sections imposes, as predicted by Riemann–Roch.\nExistence theorem (Kempf, Kleiman–Laksov): if $\\rho \\ge 0$, then $W^r_d(X)$ is nonempty for every curve $X$.\nGriffiths–Harris theorem: for a general curve $X$ of genus $g$, $W^r_d(X)$ is empty if $\\rho \u003c 0$, and it has dimension $\\rho$ if $0 \\le \\rho \\le g$.\nExample: for $g = 3$, $r = 1$, $d = 2$, one gets $\\rho = 3 - 2 \\cdot 2 = -1$, so a general curve of genus $3$ has no pencil of degree $2$ and is not hyperelliptic.\nExample: for $r = 0$ and $0 \\le d \\le g$, one gets $\\rho = d$, and $W^0_d(X)$ is the image of the $d$-fold symmetric product of $X$, of dimension $d$.\n",
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