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  "body": "\n## Summary\n\nThe moduli space $M_g$ is a geometric space whose points are the isomorphism classes of smooth projective algebraic curves of genus $g$.\nOver $\\C$ these are the compact [[Riemann surfaces|5K0N8QMV6Q]] of genus $g$, and $M_g$ is the quotient of [[Teichmüller space|ZZGC6BZNJ5]] by the [[mapping class group|JV5NYG913V]].\nFor $g \\ge 2$, $M_g$ is an irreducible quasi-projective variety of dimension $3g - 3$, and it has a compactification $\\overline{M}_g$ by stable curves due to Deligne and Mumford.\nIt is a central object of algebraic geometry, linking geometry, topology and number theory.\n\n## Explanations\n\n### L1 human\n\nEvery compact Riemann surface has a genus, the number of holes.\nSurfaces of the same genus can still differ in shape, and the moduli space is the space of all these shapes.\nFor genus $0$ there is only one shape, the Riemann sphere, so $M_0$ is a point.\nFor genus $1$ the shapes are the tori $\\C/\\Lambda$, and the $j$-invariant identifies $M_1$ with the complex line.\nFor genus $g \\ge 2$ one needs $3g - 3$ complex numbers to describe the shape, as Riemann already counted.\n",
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