55.15 Semistable reduction in genus zero
In this section we prove the semistable reduction theorem (Theorem 55.18.1) for genus zero curves.
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. If the genus of C is 0, then C is isomorphic to a conic, see Algebraic Curves, Lemma 53.10.3. Thus there exists a finite separable extension K'/K of degree at most 2 such that C(K') \not= \emptyset , see Algebraic Curves, Lemma 53.9.4. Let R' \subset K' be the integral closure of R, see discussion in More on Algebra, Remark 15.111.6. We will show that C_{K'} has semistable reduction over R'_{\mathfrak m} for each maximal ideal \mathfrak m of R' (of course in the current case there are at most two such ideals). After replacing R by R'_{\mathfrak m} and C by C_{K'} we reduce to the case discussed in the next paragraph.
In this paragraph 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, of genus 0, and C has a K-rational point. In this case C \cong \mathbf{P}^1_ K by Algebraic Curves, Proposition 53.10.4. Thus we can use \mathbf{P}^1_ R as a model and we see that C has both good and semistable reduction.
Example 55.15.1. Let R = \mathbf{R}[[\pi ]] and consider the scheme
X = V(T_1^2 + T_2^2 - \pi T_0^2) \subset \mathbf{P}^2_ R
The base change of X to \mathbf{C}[[\pi ]] is isomorphic to the scheme defined in Example 55.10.3 because we have the factorization T_1^2 + T_2^2 = (T_1 + iT_2)(T_1 - iT_2) over \mathbf{C}. Thus X is regular and its special fibre is irreducible yet singular, hence X is the unique minimal model of its generic fibre (use Lemma 55.12.4). It follows that an extension is needed even in genus 0.
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