The Stacks project

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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View Example 55.15.1 as pdf