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  "body": "\n## Statement\n\nLet $R$ be a [[discrete valuation ring|4JF1SGYZ5Y]] with field of fractions $K$.\nLet $C$ be a smooth projective geometrically connected [[curve|XJM2MT4S3Y]] over $K$ of [[genus|6W8TDV9VVN]] $g \\ge 2$.\nThen there are a finite separable extension $K'$ of $K$ and a discrete valuation ring $R' \\subset K'$ with $R' \\cap K = R$.\nMoreover there is a stable curve $\\mathcal{C}'$ over $R'$ whose generic fibre is isomorphic to $C \\times_K K'$.\n\nHere a *stable curve* over $R'$ is a [[proper|9537RKBZPP]] flat morphism $\\mathcal{C}' \\to \\operatorname{Spec} R'$ with the following properties.\nIts geometric fibres are connected reduced curves of [[arithmetic genus|H03GKRVBJ1]] $g$ whose only singularities are ordinary double points.\nEach smooth rational component of a fibre meets the other components in at least $3$ points.\n\n## Remarks\n\nThe theorem is due to Deligne and Mumford; one of their proofs deduces it from the semistable reduction of the [[Jacobian|ZQ84M9D9TC]] of $C$.\nThe stable model $\\mathcal{C}'$ is unique up to isomorphism.\nIn terms of [[moduli of curves|NK9ATH4WB6]], it says that the moduli space $\\overline{M}_g$ of stable curves satisfies the existence part of the valuative criterion of properness.\nThe extension of $K$ cannot be dropped in general.\nThe analogue for elliptic curves shows this: over $K = \\Q_p$ with $p \\ge 5$, the curve $y^2 = x^3 + p$ has additive reduction, so it has no semistable model over $\\Z_p$.\nOver $K' = \\Q_p(\\pi)$ with $\\pi^6 = p$, the substitution $x = \\pi^2 x'$, $y = \\pi^3 y'$ gives $y'^2 = x'^3 + 1$, which has good reduction.\n\n## Explanations\n\n### L1 human\n\nTake a family of smooth curves that degenerates at one point, like a family over a small disc whose central member is singular.\nThe theorem says that after a finite change of the parameter, one can replace the bad central curve by a mild one.\nThe mild curve may have several pieces, but they only cross each other in simple nodes.\nSo degenerations of smooth curves can always be tamed.\n\n### L2 human\n\nThe theorem is the key input in the proof that $\\overline{M}_g$ is proper.\nThe base change $K'/K$ is the arithmetic form of taking a root $t = s^n$ of the parameter.\nFor abelian varieties, the analogue is Grothendieck's semistable reduction theorem.\nAfter a finite extension, the identity component of the special fibre of the [[Néron model|K0EQTBRPE3]] is an extension of an abelian variety by a torus.\n",
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