55.12 Blowing down exceptional curves
The following lemma tells us what happens with the intersection numbers when we contract an exceptional curve of the first kind in a regular proper model. We put this here mostly to compare with the numerical contractions introduced in Lemma 55.3.9. We will compare the geometric and numerical contractions in Remark 55.12.3.
Lemma 55.12.1. In Situation 55.9.3 assume that $C_ n$ is an exceptional curve of the first kind. Let $f : X \to X'$ be the contraction of $C_ n$. Let $C'_ i = f(C_ i)$. Write $X'_ k = \sum m'_ i C'_ i$. Then $X'$, $C'_ i$, $i = 1, \ldots , n' = n - 1$, and $m'_ i = m_ i$ is as in Situation 55.9.3 and we have
for $i, j < n$ we have $(C'_ i \cdot C'_ j) = (C_ i \cdot C_ j) - (C_ i \cdot C_ n) (C_ j \cdot C_ n) /(C_ n \cdot C_ n)$,
for $i < n$ if $C_ i \cap C_ n \not= \emptyset $, then there are maps $\kappa _ i \leftarrow \kappa '_ i \rightarrow \kappa _ n$.
Here $\kappa _ i = H^0(C_ i, \mathcal{O}_{C_ i})$ and $\kappa '_ i = H^0(C'_ i, \mathcal{O}_{C'_ i})$.
Proof.
By Resolution of Surfaces, Lemma 54.16.8 we can contract $C_ n$ by a morphism $f : X \to X'$ such that $X'$ is regular and is projective over $R$. Thus we see that $X'$ is as in Situation 55.9.3. Let $x \in X'$ be the image of $C_ n$. Since $f$ defines an isomorphism $X \setminus C_ n \to X' \setminus \{ x\} $ it is clear that $m'_ i = m_ i$ for $i < n$.
Part (2) of the lemma is immediately clear from the existence of the morphisms $C_ i \to C'_ i$ and $C_ n \to x \to C'_ i$.
By Divisors, Lemma 31.32.11 the pullback $f^{-1}C'_ i$ is defined. By Divisors, Lemma 31.15.11 we see that $f^{-1}C'_ i = C_ i + e_ i C_ n$ for some $e_ i \geq 0$. Since $\mathcal{O}_ X(C_ i + e_ i C_ n) = \mathcal{O}_ X(f^{-1}C'_ i) = f^*\mathcal{O}_{X'}(C'_ i)$ (Divisors, Lemma 31.14.5) and since the pullback of an invertible sheaf restricts to the trivial invertible sheaf on $C_ n$ we see that
\[ 0 = \deg _{C_ n}(\mathcal{O}_ X(C_ i + e_ i C_ n)) = (C_ i + e_ i C_ n \cdot C_ n) = (C_ i \cdot C_ n) + e_ i(C_ n \cdot C_ n) \]
As $f_ j = f|_{C_ j} : C_ j \to C_ j$ is a proper birational morphism of proper curves over $k$, we see that $\deg _{C'_ j}(\mathcal{O}_{X'}(C'_ i)|_{C'_ j})$ is the same as $\deg _{C_ j}(f_ j^*\mathcal{O}_{X'}(C'_ i)|_{C'_ j})$ (Varieties, Lemma 33.44.4). Looking at the commutative diagram
\[ \xymatrix{ C_ j \ar[r] \ar[d]_{f_ j} & X \ar[d]^ f \\ C'_ j \ar[r] & X' } \]
and using Divisors, Lemma 31.14.5 we see that
\[ (C'_ i \cdot C'_ j) = \deg _{C'_ j}(\mathcal{O}_{X'}(C'_ i)|_{C'_ j}) = \deg _{C_ j}(\mathcal{O}_ X(C_ i + e_ i C_ n)) = (C_ i + e_ i C_ n \cdot C_ j) \]
Plugging in the formula for $e_ i$ found above we see that (1) holds.
$\square$
Lemma 55.12.4. Let $C$ be a smooth projective curve over $K$ with $H^0(C, \mathcal{O}_ C) = K$ and genus $0$. If there is more than one minimal model for $C$, then the special fibre of every minimal model is isomorphic to $\mathbf{P}^1_ k$.
This lemma can be improved to say that the birational transformation between two nonisomorphic minimal models can be factored as a sequence of elementary transformations as in Example 55.10.3. If we ever need this, we will precisely formulate and prove this here.
Proof.
Let $X$ be some minimal model of $C$. The numerical type associated to $X$ has genus $0$ and is minimal (Definition 55.11.4 and Lemma 55.11.5). Hence by Lemma 55.6.1 we see that $X_ k$ is reduced, irreducible, has $H^0(X_ k, \mathcal{O}_{X_ k}) = k$, and has genus $0$. Let $Y$ be a second minimal model for $C$ which is not isomorphic to $X$. 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. We will prove the lemma by induction on $m$. The base case is $m = 0$; it is true in this case because we assumed that $Y$ is minimal hence this would mean $n = 0$, but $X$ is not isomorphic to $Y$, so this does not happen, i.e., there is nothing to check.
Before we continue, note that $n + 1 = m + 1$ is equal to the number of irreducible components of the special fibre of $X_ n = Y_ m$ because both $X_ k$ and $Y_ k$ are irreducible. Another observation we will use below is that if $X' \to X''$ is a morphism of regular proper models for $C$, then $X' \to X''$ is an isomorphism over an open set of $X''$ whose complement is a finite set of closed points of the special fibre of $X''$, see Varieties, Lemma 33.17.3. In fact, any such $X' \to X''$ is a sequence of blowing ups in closed points (Resolution of Surfaces, Lemma 54.17.1) and the number of blowups is the difference in the number of irreducible components of the special fibres of $X'$ and $X''$.
Let $E_ i \subset Y_ i$, $m \geq i \geq 1$ be the curve which is contracted by the morphism $Y_ i \to Y_{i - 1}$. Let $i$ be the biggest index such that $E_ i$ has multiplicity $> 1$ in the special fibre of $Y_ i$. Then the further blowups $Y_ m \to \ldots \to Y_{i + 1} \to Y_ i$ are isomorphisms over $E_ i$ since otherwise $E_ j$ for some $j > i$ would have multiplicity $> 1$. Let $E \subset Y_ m$ be the inverse image of $E_ i$. By what we just said $E \subset Y_ m$ is an exceptional curve of the first kind. Let $Y_ m \to Y'$ be the contraction of $E$ (which exists by Resolution of Surfaces, Lemma 54.16.9). The morphism $Y_ m \to X$ has to contract $E$, because $X_ k$ is reduced. Hence there are morphisms $Y' \to Y$ and $Y' \to X$ (by Resolution of Surfaces, Lemma 54.16.1) which are compositions of at most $n - 1 = m - 1$ contractions of exceptional curves (see discussion above). We win by induction on $m$. Upshot: we may assume that the special fibres of all of the curves $X_ i$ and $Y_ i$ are reduced.
Since the fibres of $X_ i$ and $Y_ i$ are reduced, it has to be the case that the blowups $X_ i \to X_{i - 1}$ and $Y_ i \to Y_{i - 1}$ happen in closed points which are regular points of the special fibres. Namely, if $X''$ is a regular model for $C$ and if $x \in X''$ is a closed point of the special fibre, and $\pi \in \mathfrak m_ x^2$, then the exceptional fibre $E$ of the blowup $X' \to X''$ at $x$ has multiplicity at least $2$ in the special fibre of $X'$ (local computation omitted). Hence $\mathcal{O}_{X''_ k, x} = \mathcal{O}_{X'', x}/\pi $ is regular (Algebra, Lemma 10.106.3) as claimed. In particular $x$ is a Cartier divisor on the unique irreducible component $Z'$ of $X''_ k$ it lies on (Varieties, Lemma 33.43.8). It follows that the strict transform $Z \subset X'$ of $Z'$ maps isomorphically to $Z'$ (use Divisors, Lemmas 31.33.2 and 31.32.7). In other words, if an irreducible component $Z$ of $X_ i$ is not contracted under the map $X_ i \to X_ j$ ($i > j$) then it maps isomorphically to its image.
Now we are ready to prove the lemma. Let $E \subset Y_ m$ be the exceptional curve of the first kind which is contracted by the morphism $Y_ m \to Y_{m - 1}$. If $E$ is contracted by the morphism $Y_ m = X_ n \to X$, then there is a factorization $Y_{m - 1} \to X$ (Resolution of Surfaces, Lemma 54.16.1) and moreover $Y_{m - 1} \to X$ is a sequence of blowups in closed points (Resolution of Surfaces, Lemma 54.17.1). In this case we lower $m$ and we win by induction. Finally, assume that $E$ is not contracted by the morphism $Y_ m \to X$. Then $E \to X_ k$ is surjective as $X_ k$ is irreducible and by the above this means it is an isomorphism. Hence $X_ k$ is isomorphic to a projective line as desired.
$\square$
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