Lemma 55.10.1. Let C be a smooth projective curve over K with H^0(C, \mathcal{O}_ C) = K and genus > 0. There is a unique minimal model for C.
55.10 Uniqueness of the minimal model
If the genus of the generic fibre is positive, then minimal models are unique (Lemma 55.10.1) and consequently have a suitable mapping property (Lemma 55.10.2).
Proof. We have already proven the hard part of the lemma which is the existence of a minimal model (whose proof relies on resolution of surface singularities), see Proposition 55.8.6. To prove uniqueness, suppose that X and Y are two minimal models. By Resolution of Surfaces, Lemma 54.17.2 there exists a diagram of S-morphisms
X = X_0 \leftarrow X_1 \leftarrow \ldots \leftarrow X_ n = Y_ m \to \ldots \to Y_1 \to Y_0 = Y
where each morphism is a blowup in a closed point. The exceptional fibre of the morphism X_ n \to X_{n - 1} is an exceptional curve of the first kind E. We claim that E is contracted to a point under the morphism X_ n = Y_ m \to Y. If this is true, then X_ n \to Y factors through X_{n - 1} by Resolution of Surfaces, Lemma 54.16.1. In this case the morphism X_{n - 1} \to Y is still a sequence of contractions of exceptional curves by Resolution of Surfaces, Lemma 54.17.1. Hence by induction on n we conclude. (The base case n = 0 means that there is a sequence of contractions X = Y_ m \to \ldots \to Y_1 \to Y_0 = Y ending with Y. However as X is a minimal model it contains no exceptional curves of the first kind, hence m = 0 and X = Y.)
Proof of the claim. We will show by induction on m that any exceptional curve of the first kind E \subset Y_ m is mapped to a point by the morphism Y_ m \to Y. If m = 0 this is clear because Y is a minimal model. If m > 0, then either Y_ m \to Y_{m - 1} contracts E (and we're done) or the exceptional fibre E' \subset Y_ m of Y_ m \to Y_{m - 1} is a second exceptional curve of the first kind. Since both E and E' are irreducible components of the special fibre and since g_ C > 0 by assumption, we conclude that E \cap E' = \emptyset by Lemma 55.9.11. Then the image of E in Y_{m - 1} is an exceptional curve of the first kind (this is clear because the morphism Y_ m \to Y_{m - 1} is an isomorphism in a neighbourhood of E). By induction we see that Y_{m - 1} \to Y contracts this curve and the proof is complete. \square
Lemma 55.10.2. Let C be a smooth projective curve over K with H^0(C, \mathcal{O}_ C) = K and genus > 0. Let X be the minimal model for C (Lemma 55.10.1). Let Y be a regular proper model for C. Then there is a unique morphism of models Y \to X which is a sequence of contractions of exceptional curves of the first kind.
Proof. The existence and properties of the morphism X \to Y follows immediately from Lemma 55.8.5 and the uniqueness of the minimal model. The morphism Y \to X is unique because C \subset Y is scheme theoretically dense and X is separated (see Morphisms, Lemma 29.7.10). \square
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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