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  "body": "\n## Summary\n\nBrill–Noether theory studies the linear systems on an [[algebraic curve|XJM2MT4S3Y]].\nFor a given degree $d$ and dimension $r$, it asks which line bundles of degree $d$ have at least $r + 1$ independent sections.\nThe answer for a general curve of genus $g$ is governed by the Brill–Noether number $\\rho = g - (r + 1)(g - d + r)$.\nSuch line bundles exist on every curve when $\\rho \\ge 0$, and they do not exist on a general curve when $\\rho \u003c 0$.\nThe theory organizes the maps of curves to projective spaces and is a central tool in the study of moduli of curves.\n\n## Explanations\n\n### L1 human\n\nA curve can be mapped to a line, a plane or a higher projective space in many ways, and each map comes from a line bundle with enough sections.\nBrill–Noether theory asks which maps of a given degree into a given space exist.\nA simple count of conditions predicts the answer, and for a typical curve the prediction is right.\nFor instance, the count says that a typical curve of genus $3$ has no map of degree $2$ to the line, so it is not hyperelliptic.\n",
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