Semistable reduction theorem

Remarks

The theorem is due to Deligne and Mumford; one of their proofs deduces it from the semistable reduction of the Jacobian of $C$. The stable model $\mathcal{C}'$ is unique up to isomorphism. In terms of moduli of curves, it says that the moduli space $\overline{M}_g$ of stable curves satisfies the existence part of the valuative criterion of properness. The extension of $K$ cannot be dropped in general. The 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 $\Z_p$. Over $K' = \Q_p(\pi)$ with $\pi^6 = p$, the substitution $x = \pi^2 x'$, $y = \pi^3 y'$ gives $y'^2 = x'^3 + 1$, which has good reduction.

Going deeper

For a student

The theorem is the key input in the proof that $\overline{M}_g$ is proper. The base change $K'/K$ is the arithmetic form of taking a root $t = s^n$ of the parameter. For abelian varieties, the analogue is Grothendieck's semistable reduction theorem. After a finite extension, the identity component of the special fibre of the Néron model is an extension of an abelian variety by a torus.