The Stacks project

Lemma 55.11.5. In Situation 55.9.3. The following are equivalent

  1. $X$ is a minimal model, and

  2. the numerical type associated to $X$ is minimal.

Proof. If the numerical type is minimal, then there is no $i$ with $g_ i = 0$ and $(C_ i \cdot C_ i) = -[\kappa _ i: k]$, see Definition 55.3.12. Certainly, this implies that none of the curves $C_ i$ are exceptional curves of the first kind.

Conversely, suppose that the numerical type is not minimal. Then there exists an $i$ such that $g_ i = 0$ and $(C_ i \cdot C_ i) = -[\kappa _ i: k]$. We claim this implies that $C_ i$ is an exceptional curve of the first kind. Namely, the invertible sheaf $\mathcal{O}_ X(-C_ i)|_{C_ i}$ has degree $-(C_ i \cdot C_ i) = [\kappa _ i : k]$ when $C_ i$ is viewed as a proper curve over $k$, hence has degree $1$ when $C_ i$ is viewed as a proper curve over $\kappa _ i$. Applying Algebraic Curves, Proposition 53.10.4 we conclude that $C_ i \cong \mathbf{P}^1_{\kappa _ i}$ as schemes over $\kappa _ i$. Since the Picard group of $\mathbf{P}^1$ over a field is $\mathbf{Z}$, we see that the normal sheaf of $C_ i$ in $X$ is isomorphic to $\mathcal{O}_{\mathbf{P}_{\kappa _ i}}(-1)$ and the proof is complete. $\square$

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