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

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 let $Z \subseteq X$ be a nonzero effective Cartier divisor with $Z \subseteq f^{- 1} (y)$ set-theoretically. Then there exists an irreducible component $C$ of $Z$ with $\deg_{C} (O_{X} (- Z) ∣_{C}) > 0$ ; that is, the conormal sheaf $O_{X} (- Z) ∣_{Z}$ has a component of positive degree.

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 one-dimensional fibre components $C_{1}, \dots, C_{r}$ , and a nonzero effective Cartier divisor $Z \subseteq f^{- 1} (y)$ supported set-theoretically in the fibre.

[F1]

def-associated-prime-of-a-module. Let $R$ be a commutative ring and let $M$ be a left $R$ -module. A prime ideal $p ⊊ R$ is associated to $M$ when $p = {Ann}_{R} (m)$ for some element $m \in M$ . The set of associated primes of $M$ is denoted ${Ass}_{R} (M) .$ (Associated primes of a module)

[F2]

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)

[F3]

def-degree-invertible-sheaf-proper-dimension-one. Assume the Axiom of Choice, inherited from the Euler-characteristic supplier below (The Axiom of Choice). Let $k$ be a field (def-field) and let $C$ be a proper $k$ -scheme (def-proper-morphism) whose underlying topological space is Noetherian of dimension at most one (def-dimension-noetherian-topological-space, def-locally-noetherian-and-noetherian-scheme). (Degree of an invertible sheaf on a proper one-dimensional scheme)

[F4]

def-effective-cartier-divisor. A Cartier divisor $D$ on a scheme $X$ is effective if it has a local-equation representation $(U_{i}, f_{i})$ as in def-cartier-divisor with $f_{i} \in O_{X} (U_{i})$ and with multiplication by the germ $(f_{i})_{x}$ injective on $O_{X, x}$ for every $x \in U_{i}$ . (Effective cartier divisor)

[F5]

def-invertible-sheaf-of-cartier-divisor. Let $D$ be a Cartier divisor on a scheme $X$ , represented by meromorphic units $f_{i} \in K_{X} (U_{i})^{\times}$ with regular-unit ratios on the overlaps (def-cartier-divisor). The subsheaf $O_{X} \subseteq K_{X}$ is the one of def-sheaf-total-quotient-rings. (Invertible sheaf of cartier divisor)

[F6]

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)

[F7]

lem-effective-cartier-divisor-has-no-embedded-associated-primes. Assume the Axiom of Choice (The Axiom of Choice). Let $X$ be a regular locally Noetherian scheme (A local ring is a nonzero commutative ring with a unique maximal ideal) and let $Z \subseteq X$ be an effective Cartier divisor, with associated closed subscheme $Z = Z_{D} ↪ X$ and ideal sheaf $I_{D}$ (Effective cartier divisor, thm-effective-cartier-divisor-closed-immersion). (Effective Cartier divisors on a regular scheme have no embedded associated points)

[F8]

lem-existence-of-a-fibre-cutter. 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 $C_{1}, \dots, C_{r}$ . (A function cutting the components of a special fibre)

[F9]

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)

[F10]

lem-nonzero-section-vanishing-at-a-point-has-positive-degree. Assume the Axiom of Choice. Let $k$ be a field, let $C$ be an integral proper $k$ -scheme of dimension one and let $L$ be an invertible $O_{C}$ -module with a nonzero global section $s \in Γ (C, L)$ . If $s$ vanishes at some closed point of $C$ , then $\deg_{C} (L) > 0$ for the degree of Degree of an invertible sheaf on a proper one-dimensional scheme. (A nonzero section vanishing at a point forces positive degree)

[F11]

thm-one-dimensional-regular-local-rings-are-dvrs. Assume the Axiom of Choice (The Axiom of Choice). A nonzero Noetherian local ring of dimension one is regular if and only if it is a discrete valuation ring. Fields are excluded from the term DVR. (one dimensional regular local rings are dvrs)

[F12]

cor-degree-additive-proper-curve. Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field and let $C$ be a proper $k$ -scheme (def-proper-morphism) whose underlying topological space has dimension at most one (def-dimension-noetherian-topological-space). For all invertible $O_{C}$ -modules $L$ and $M$ (def-invertible-sheaf): 1. (Degree is additive on invertible sheaves over a proper curve)

Proof

1.1F8F9F11given

Write $Z = \sum_{j} d_{j} C_{j}$ with $d_{j} \geq 0$ , not all zero, and use the fibre cutter $u$ with orders $e_{j} = v_{C_{j}} (u) > 0$ at all components; choose an index $i$ with $d_{i} / e_{i}$ maximal and replace $u$ by $u^{d_{i}}$ and $Z$ by $e_{i} Z$ , so that $v_{C_{j}} (u) \geq v_{C_{j}} (Z)$ for every $j$ with equality at $i$ .

2.1F1F4F5F7step 1.1

The order inequalities say that $u$ vanishes at the generic point of every component of $Z$ , hence at every associated point of the effective Cartier divisor $Z$ ; since $Z$ has no embedded associated points, the section $u$ is zero on $Z$ . Thus $u$ lies in the ideal $I = O_{X} (- Z)$ , and its image in $I ∣_{C}$ restricted to $C_{i}$ is nonzero at the generic point, because the order at $C_{i}$ is exactly $v_{C_{i}} (Z)$ .

3.1F6F7F8step 2.1

Near the chosen point $x_{i} \in C_{i}$ with $x_{i} \notin C_{j}$ for $j \neq i$ , choose a local generator $t$ of $I$ ; the divisor $Z$ has only the component $C_{i}$ there, and $g_{i}$ is nonzero on $C_{i}$ while $Z$ has no embedded associated points, so multiplication by $g_{i}$ on $O_{Z}$ is injective.

4.1F6F7step 3.1

From $u = g_{i} h_{i} \in (t)$ and the vanishing of $u$ on $Z$ we get $h_{i} \in (t)$ : the image of $h_{i}$ in $O_{Z}$ is killed by the injective multiplication by $g_{i}$ and hence is zero. Therefore the section coefficient $u / t = g_{i} \cdot (h_{i} / t)$ vanishes at $x_{i}$ .

5.1F3F10F12step 1.1step 4.1F2∎

The section $u / t$ of the invertible sheaf $I ∣_{C_{i}}$ is nonzero and vanishes at the closed point $x_{i}$ , so the positive-degree lemma gives $\deg_{C_{i}} (I ∣_{C_{i}}) > 0$ for the rescaled data; by additivity of the degree under tensor powers, dividing by the positive scaling factor $e_{i}$ returns the same positivity for the original conormal sheaf, proving the claim.

Remarks

The rescaling by d_i/e_i is what makes the section vanish on all of Z while remaining nonzero on the chosen component.
No embedded associated points of Z is used twice: to conclude that u vanishes on Z and that multiplication by g_i is injective on O_Z.

Depends on

Used by

Negativity of contracted curves on regular surfaces Theorem

Dependency tree · two levels

74 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)