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  "body": "\n## Definition\n\nLet $g \\ge 0$ be an integer.\nLet $T$ be a scheme over $\\C$.\nA *family of curves of [[genus|NJ787GZS1N]] $g$* over $T$ is a smooth proper morphism $\\pi \\colon C \\to T$.\nIts fibres must be connected algebraic curves of genus $g$.\nLet $F_g$ be the functor that sends $T$ to the set of isomorphism classes of families of curves of genus $g$ over $T$.\nIt sends a morphism $T' \\to T$ to the pullback of families.\nA *coarse moduli space of curves of genus $g$* is a scheme $M_g$ over $\\C$ with a natural transformation $\\Phi \\colon F_g \\to \\Hom(-, M_g)$ such that:\n\n1. $\\Phi$ is a bijection from $F_g(\\operatorname{Spec} \\C)$ onto the set of $\\C$-points of $M_g$, and\n2. for every scheme $N$ and natural transformation $\\Psi \\colon F_g \\to \\Hom(-, N)$ there is a unique morphism $u \\colon M_g \\to N$ with $\\Psi = u \\circ \\Phi$.\n\n## Remarks\n\nHere $F_g(\\operatorname{Spec} \\C)$ is the set of isomorphism classes of smooth projective connected curves of genus $g$.\nA coarse moduli space is unique up to unique isomorphism, and it exists for every $g$.\nIt is not a fine moduli space, that is, $\\Phi$ is not an isomorphism of functors, because some curves have nontrivial automorphisms.\nThe analytic space underlying $M_g$ is the quotient of the [[Teichmüller space|ZZGC6BZNJ5]] $T_g$ by the mapping class group.\n\nExample: $g = 1$.\nEach curve of genus $1$ over $\\C$ is an elliptic curve, and the $j$-invariant gives a bijection from isomorphism classes to $\\C$.\nThis bijection comes from a natural transformation as in the definition, and $M_1$ is the affine line $\\mathbb{A}^1$.\n\nNon-example: the bare set $F_g(\\operatorname{Spec} \\C)$ has no scheme structure.\nSo it is not a coarse moduli space by itself, although it is in bijection with the points of $M_g$.\n",
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