The Stacks project

55.17 Semistable reduction in genus at least two

In this section we prove the semistable reduction theorem (Theorem 55.18.1) for curves of genus $\geq 2$. Fix $g \geq 2$.

Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_ C) = K$. Assume the genus of $C$ is $g$. Choose a prime $\ell > 768g$ different from the characteristic of $k$. Choose a finite separable extension $K'/K$ of such that $C(K') \not= \emptyset $ and such that $\mathop{\mathrm{Pic}}\nolimits (C_{K'})[\ell ] \cong (\mathbf{Z}/\ell \mathbf{Z})^{\oplus 2g}$. See Algebraic Curves, Lemma 53.17.2. Let $R' \subset K'$ be the integral closure of $R$, see discussion in More on Algebra, Remark 15.111.6. We may replace $R$ by $R'_{\mathfrak m}$ for some maximal ideal $\mathfrak m$ in $R'$ and $C$ by $C_{K'}$. This reduces us to the case discussed in the next paragraph.

In the rest of this section $R$ is a discrete valuation ring with fraction field $K$, $C$ is a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_ C) = K$, with genus $g$, having a $K$-rational point, and with $\mathop{\mathrm{Pic}}\nolimits (C)[\ell ] \cong (\mathbf{Z}/\ell \mathbf{Z})^{\oplus 2g}$ for some prime $\ell \geq 768g$ different from the characteristic of $k$. We will prove that $C$ has semistable reduction.

In the rest of this section we will use without further mention that the conclusions of Lemma 55.11.7 are true.

Let $X$ be a minimal model for $C$, see Proposition 55.8.6. Let $T = (n, m_ i, (a_{ij}), w_ i, g_ i)$ be the numerical type associated to $X$ (Definition 55.11.4). Then $T$ is a minimal numerical type of genus $g$ (Lemma 55.11.5). By Proposition 55.7.4 we have

By Lemmas 55.13.3 and 55.13.4 we conclude that there is an embedding

By Algebraic Curves, Lemma 53.18.6 we obtain

By Lemmas 55.11.8 and 55.11.9 we have

Elementary number theory tells us that the only way these $3$ inequalities can hold is if they are all equalities. Looking at Lemma 55.11.8 we conclude that $m_ i = 1$ for all $i$. Looking at Lemma 55.11.10 we conclude that every irreducible component of $X_ k$ is smooth over $k$.

In particular, since $X_ k$ is the scheme theoretic union of its irreducible components $C_ i$ we see that $X_{\overline{k}}$ is the scheme theoretic union of the $C_{i, \overline{k}}$. Hence $X_{\overline{k}}$ is a reduced connected proper scheme of dimension $1$ over $\overline{k}$ with $\dim _{\overline{k}} H^1(X_{\overline{k}}, \mathcal{O}_{X_{\overline{k}}}) = g$. Also, by Varieties, Lemma 33.30.3 and the above we still have

By Algebraic Curves, Proposition 53.17.3 we see that $X_{\overline{k}}$ has at only multicross singularities. But since $X_ k$ is Gorenstein (Lemma 55.9.2), so is $X_{\overline{k}}$ (Duality for Schemes, Lemma 48.25.1). We conclude $X_{\overline{k}}$ is at-worst-nodal by Algebraic Curves, Lemma 53.16.4. This finishes the proof.