The Stacks project

Example 55.10.3. If the genus of $C$ is $0$, then minimal models are indeed nonunique. Namely, consider the closed subscheme

\[ X \subset \mathbf{P}^2_ R \]

defined by $T_1T_2 - \pi T_0^2 = 0$. More precisely $X$ is defined as $\text{Proj}(R[T_0, T_1, T_2]/(T_1T_2 - \pi T_0^2))$. Then the special fibre $X_ k$ is a union of two exceptional curves $C_1$, $C_2$ both isomorphic to $\mathbf{P}^1_ k$ (exactly as in Lemma 55.9.11). Projection from $(0 : 1 : 0)$ defines a morphism $X \to \mathbf{P}^1_ R$ contracting $C_2$ and inducing an isomorphism of $C_1$ with the special fiber of $\mathbf{P}^1_ R$. Projection from $(0 : 0 : 1)$ defines a morphism $X \to \mathbf{P}^1_ R$ contracting $C_1$ and inducing an isomorphism of $C_2$ with the special fiber of $\mathbf{P}^1_ R$. More precisely, these morphisms correspond to the graded $R$-algebra maps

\[ R[T_0, T_1] \longrightarrow R[T_0, T_1, T_2]/(T_1T_2 - \pi T_0^2) \longleftarrow R[T_0, T_2] \]

In Lemma 55.12.4 we will study this phenomenon.

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View Example 55.10.3 as pdf