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54.18 Semistable reduction for curves

In this section we finish the proof of the theorem. For $g \geq 2$ let $768g < \ell ' < \ell $ be the first two primes $> 728g$ and set

54.18.0.1
\begin{equation} \label{models-equation-bound} B_ g = (2g - 2)(\ell ^{2g})! \end{equation}

The precise form of $B_ g$ is unimportant; the point we are trying to make is that it depends only on $g$.

reference

Theorem 54.18.1. Let $R$ be a discrete valuation ring with fraction field $K$. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_ C) = K$. Then there exists an extension of discrete valuation rings $R \subset R'$ which induces a finite separable extension of fraction fields $K \subset K'$ such that $C_{K'}$ has semistable reduction. More precisely, we have the following

  1. If the genus of $C$ is zero, then there exists a degree $2$ separable extension $K'/K$ such that $C_{K'} \cong \mathbf{P}^1_{K'}$ and hence $C_{K'}$ is isomorphic to the generic fibre of the smooth projective scheme $\mathbf{P}^1_{R'}$ over the integral closure $R'$ of $R$ in $K'$.

  2. If the genus of $C$ is one, then there exists a finite separable extension $K'/K$ such that $C_{K'}$ has semistable reduction over $R'_\mathfrak m$ for every maximal ideal $\mathfrak m$ of the integral closure $R'$ of $R$ in $K'$. Moreover, the special fibre of the (unique) minimal model of $C_{K'}$ over $R'_\mathfrak m$ is either a smooth genus one curve or a cycle of rational curves.

  3. If the genus $g$ of $C$ is greater than one, then there exists a finite separable extension $K'/K$ of degree at most $B_ g$ (54.18.0.1) such that $C_{K'}$ has semistable reduction over $R'_\mathfrak m$ for every maximal ideal $\mathfrak m$ of the integral closure $R'$ of $R$ in $K'$.

Proof. For the case of genus zero, see Section 54.15. For the case of genus one, see Section 54.16. For the case of genus greater than one, see Section 54.17. To see that we have a bound on the degree $[K' : K]$ you can use the bound on the degree of the extension needed to make all $\ell $ or $\ell '$ torsion visible proved in Algebraic Curves, Lemma 52.17.2. (The reason for using $\ell $ and $\ell '$ is that we need to avoid the characteristic of the residue field $k$.) $\square$

Remark 54.18.2 (Improving the bound). Results in the literature suggest that one can improve the bound given in the statement of Theorem 54.18.1. For example, in [DM] it is shown that semistable reduction of $C$ and its Jacobian are the same thing if the residue field is perfect and presumably this is true for general residue fields as well. For an abelian variety we have semistable reduction if the action of Galois on the $\ell $-torsion is trivial for any $\ell \geq 3$ not equal to the residue characteristic. Thus we can presumably choose $\ell = 5$ in the formula (54.18.0.1) for $B_ g$ (but the proof would take a lot more work; if we ever need this we will make a precise statement and provide a proof here).


Comments (1)

Comment #4530 by 李时璋 on

very very small issue, on the first line you had 768g, but on the second line you were saying 728g.


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