Lemma 54.17.1. Let $f : X \to Y$ be a proper birational morphism between integral Noetherian schemes regular of dimension $2$. Then $f$ is a sequence of blowups in closed points.
Proof. Let $V \subset Y$ be the maximal open over which $f$ is an isomorphism. Then $V$ contains all codimension $1$ points of $V$ (Varieties, Lemma 33.17.3). Let $y \in Y$ be a closed point not contained in $V$. Then we want to show that $f$ factors through the blowup $b : Y' \to Y$ of $Y$ at $y$. Namely, if this is true, then at least one (and in fact exactly one) component of the fibre $f^{-1}(y)$ will map isomorphically onto the exceptional curve in $Y'$ and the number of curves in fibres of $X \to Y'$ will be strictly less that the number of curves in fibres of $X \to Y$, so we conclude by induction. Some details omitted.
By Lemma 54.4.3 we know that there exists a sequence of blowing ups
\[ X' = X_ n \to X_{n - 1} \to \ldots \to X_1 \to X_0 = X \]
in closed points lying over the fibre $f^{-1}(y)$ and a morphism $X' \to Y'$ such that
\[ \xymatrix{ X' \ar[d]_{f'} \ar[r] & X \ar[d]^ f \\ Y' \ar[r] & Y } \]
is commutative. We want to show that the morphism $X' \to Y'$ factors through $X$ and hence we can use induction on $n$ to reduce to the case where $X' \to X$ is the blowup of $X$ in a closed point $x \in X$ mapping to $y$.
Let $E \subset X'$ be the exceptional fibre of the blowing up $X' \to X$. If $E$ maps to a point in $Y'$, then we obtain the desired factorization by Lemma 54.16.1. We will prove that if this is not the case we obtain a contradiction. Namely, if $f'(E)$ is not a point, then $E' = f'(E)$ must be the exceptional curve in $Y'$. Picture
\[ \xymatrix{ E \ar[r] \ar[d]_ g & X' \ar[d]_{f'} \ar[r] & X \ar[d]^ f \\ E' \ar[r] & Y' \ar[r] & Y } \]
Arguing as before $f'$ is an isomorphism in an open neighbourhood of the generic point of $E'$. Hence $g : E \to E'$ is a finite birational morphism. Then the inverse of $g$ (a rational map) is everywhere defined by Morphisms, Lemma 29.42.5 and $g$ is an isomorphism. Consider the map
\[ g^*\mathcal{C}_{E'/Y'} \longrightarrow \mathcal{C}_{E/X'} \]
of Morphisms, Lemma 29.31.3. Since the source and target are invertible modules of degree $1$ on $E = E' = \mathbf{P}^1_\kappa $ and since the map is nonzero (as $f'$ is an isomorphism in the generic point of $E$) we conclude it is an isomorphism. By Morphisms, Lemma 29.32.18 we conclude that $\Omega _{X'/Y'}|_ E = 0$. This means that $f'$ is unramified at every point of $E$ (Morphisms, Lemma 29.35.14). Hence $f'$ is quasi-finite at every point of $E$ (Morphisms, Lemma 29.35.10). Hence the maximal open $V' \subset Y'$ over which $f'$ is an isomorphism contains $E'$ by Varieties, Lemma 33.17.3. This in turn implies that the inverse image of $y$ in $X'$ is $E'$. Hence the inverse image of $y$ in $X$ is $x$. Hence $x \in X$ is in the maximal open over which $f$ is an isomorphism by Varieties, Lemma 33.17.3. This is a contradiction as we assumed that $y$ is not in this open. $\square$
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (0)