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A function cutting the components of a special fibre

Statement

Assume the Axiom of Choice. Let $k$ , $X$ , $Y$ , $f ⁣ : X \to Y$ and $y \in Y$ be as in Fibres of a proper birational morphism of regular surfaces, and assume $\dim f^{- 1} (y) = 1$ with components $C_{1}, \dots, C_{r}$ . Then there exists a nonzero element $u \in O_{Y, y}$ such that for every $i$ there is a closed point $x_{i} \in C_{i}$ with $x_{i} \notin C_{j}$ for $j \neq i$ and a factorization $u = g_{i} h_{i}$ in $O_{X, x_{i}}$ under the local homomorphism $f^{♯} ⁣ : O_{Y, y} \to O_{X, x_{i}}$ in which $g_{i} \in m_{X, x_{i}}$ maps to a nonzero element of $O_{C_{i}, x_{i}}$ and $h_{i} \in O_{X, x_{i}}$ .

Facts & Assumptions

Given: A field $k$ , integral regular finite-type $k$ -schemes $X, Y$ of pure dimension two, a proper birational morphism $f ⁣ : X \to Y$ , a closed point $y \in Y$ with $\dim f^{- 1} (y) = 1$ and components $C_{1}, \dots, C_{r}$ as in the fibre-components lemma.

[F1]

def-axiom-of-choice. The Axiom of Choice (AC) is the following statement. > Every family of nonempty sets has a choice function > (def-choice-function). Written out: for every set $F$ all of whose members are nonempty, there exists a function $g$ with domain $F$ satisfying $g (S) \in S$ for all $S \in F$ . (The Axiom of Choice)

[F2]

def-local-ring. A local ring is a nonzero commutative ring $R$ with exactly one maximal ideal. That ideal is usually denoted $m_{R}$ or simply $m$ . The quotient $R / m$ , which is a field, is the residue field of the local ring. (A local ring is a nonzero commutative ring with a unique maximal ideal)

[F3]

def-stalk-of-presheaf. Let $F$ be a presheaf on a topological space $X$ , and let $x \in X$ . The neighbourhood category of $x$ is the full subcategory $N_{x} \subseteq Open (X)$ whose objects are the open neighbourhoods of $x$ . (The stalk of a presheaf at a point)

[F4]

lem-fibre-components-of-a-proper-birational-morphism-of-regular-surfaces. Assume the Axiom of Choice. Let $k$ be a field, let $X$ and $Y$ be integral regular finite-type $k$ -schemes of pure dimension two, let $f ⁣ : X \to Y$ be a proper birational morphism and let $y \in Y$ be a closed point. Put $F = f^{- 1} (y)$ . Then: 1. $F$ is a proper $κ (y)$ -scheme with $\dim F \leq 1$ . 2. (Fibres of a proper birational morphism of regular surfaces)

Proof

1.1F2F3F4given

By part 2 of the fibre-components lemma choose for every $i$ a closed point $x_{i} \in C_{i}$ with $x_{i} \notin C_{j}$ for $j \neq i$ , and choose any $g_{i} \in m_{X, x_{i}}$ whose image in the local ring $O_{C_{i}, x_{i}}$ is nonzero; such an element exists because the quotient map $O_{X, x_{i}} \to O_{C_{i}, x_{i}}$ sends the maximal ideal onto the maximal ideal of the nonzero local ring $O_{C_{i}, x_{i}}$ .

2.1F4step 1.1

The local homomorphism $f^{♯} ⁣ : O_{Y, y} \to O_{X, x_{i}}$ is injective on local rings and becomes an isomorphism of fraction fields: by part 3 of the fibre-components lemma the function field of $Y$ is identified with the function field of $X$ under $f^{♯}$ , so the germ $g_{i}$ can be written as a quotient $g_{i} = a_{i} / b_{i}$ with $a_{i}, b_{i} \in O_{Y, y}$ and $b_{i} \neq 0$ .

3.1F2step 2.1

Put $u : = \prod_{j} a_{j} \in O_{Y, y}$ , a nonzero element because each $a_{j}$ is nonzero and the local ring is a domain. Each $a_{j}$ is in the target maximal ideal: otherwise its pullback would be a unit, contradicting $a_{j} = g_{j} b_{j}$ with $g_{j}$ a nonunit. Thus $u$ lies in that maximal ideal and has positive valuation along every fibre curve; then in $O_{X, x_{i}}$ one has $u = g_{i} \cdot h_{i}$ with $h_{i} : = b_{i} \prod_{j \neq i} a_{j}$ , and by construction $g_{i}$ maps to a nonzero element of $O_{C_{i}, x_{i}}$ .

4.1F1step 3.1∎

The element $u$ and the factorizations of step 3.1 are the required data; the Axiom of Choice is inherited from the cited fibre-components lemma.

Remarks

The function $u$ is a concrete product of numerators obtained from the birational identification of function fields; no glueing or approximation argument is used.
The points $x_{i}$ are closed points chosen to lie on no other component; smoothness of the components or these points is not assumed.

Depends on

Used by

A divisor supported in a special fibre has positive conormal degree on some component Lemma

Dependency tree · two levels

32 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

The Stacks Project, Resolution of Surfaces, Section 54.7 (Vanishing) (standard reference, not scraped)