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  "body": "\n## Definition\n\nLet $S$ be a base scheme, and let $F \\colon (\\mathrm{Sch}/S)^{\\mathrm{op}} \\to \\mathrm{Set}$ be a contravariant [[functor|28NQ0KRHD3]], the *moduli functor*.\nThe set $F(T)$ consists of the isomorphism classes of families of the objects to be classified, parametrized by the [[scheme|5A35TYZD02]] $T$.\nThe functor $F$ acts on morphisms by pullback of families.\nFor an $S$-scheme $M$, write $h_M = \\Hom_S(-, M)$ for its [[functor of points|SWWW9X4A3N]].\n\n1. A *fine moduli space* for $F$ is an $S$-scheme $M$ with an isomorphism of functors $F \\cong h_M$, that is, $M$ [[represents|KCCHH87H6E]] $F$.\n   The family $\\mathcal{U} \\in F(M)$ that corresponds to $\\id_M$ is the *universal family*.\n2. A *coarse moduli space* for $F$ is an $S$-scheme $M$ with a natural transformation $\\Phi \\colon F \\to h_M$ such that:\n   1. for every [[algebraically closed field|02SJG1HC32]] $\\Omega$ and morphism $\\operatorname{Spec} \\Omega \\to S$, the map $\\Phi \\colon F(\\operatorname{Spec} \\Omega) \\to M(\\Omega)$ is a bijection;\n   2. for every $S$-scheme $N$ and natural transformation $\\Psi \\colon F \\to h_N$, there is a unique morphism $f \\colon M \\to N$ with $\\Psi = h_f \\circ \\Phi$.\n\n## Remarks\n\nA fine moduli space is also a coarse one, and every family over $T$ is the pullback of $\\mathcal{U}$ along a unique map $T \\to M$.\nExample: let $F(T)$ be the set of line subbundles $L \\subset \\mathcal{O}_T^{n+1}$ with locally free quotient.\nThen $\\mathbb{P}^n_S$ is a fine moduli space for $F$, with universal family the tautological line bundle $\\mathcal{O}(-1) \\subset \\mathcal{O}^{n+1}$.\nNon-example: elliptic curves over an algebraically closed field of characteristic $0$ have no fine moduli space.\nEvery elliptic curve has the automorphism $-1$, and this gives nontrivial families whose fibers are all isomorphic.\nThe affine line, through the [[j-invariant|9538W1K1EX]], is a coarse moduli space for them.\nAutomorphisms are the usual obstruction to fine moduli spaces, and moduli stacks are a way around it.\n",
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