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Intersection with a curve is the degree of the restriction

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field, let $X$ be an integral regular projective surface over $k$ (Intersection numbers of Cartier divisors on a smooth projective surface) and let $C$ and $D$ be effective Cartier divisors on $X$ (Effective cartier divisor, Cartier divisor) with associated line bundles $O_{X} (C)$ and $O_{X} (D)$ (Invertible sheaf of cartier divisor). Then $C \cdot D = \deg_{C} (O_{X} (D) ∣_{C}) = \deg_{D} (O_{X} (C) ∣_{D}),$ where $\cdot$ is the intersection product of Intersection numbers of Cartier divisors on a smooth projective surface and $\deg$ is the degree of Degree of an invertible sheaf on a proper one-dimensional scheme on the proper curves $C$ and $D$ .

More generally, if only $C$ is assumed effective, then the first identity $C \cdot D = \deg_{C} (O_{X} (D) ∣_{C})$ holds for every Cartier divisor $D$ on $X$ , with $\deg_{C}$ taken on the curve $C$ . If $C$ is moreover a smooth proper geometrically integral curve, then $\deg_{C}$ agrees with the closed-point divisor degree of Degree divisor proper curve on $C$ . The empty curve case $C = \emptyset$ is included: both sides are then $0$ .

Facts & Assumptions

Given: a field $k$ , an integral regular projective surface $X$ over $k$ , an effective Cartier divisor $C \subseteq X$ , and a Cartier divisor $D$ on $X$ .

[F1]

Effective Cartier divisors and closed immersions: an effective Cartier divisor $C$ is given by local equations that are nonzerodivisors, its ideal sheaf $I_{C} = O_{X} (- C)$ is invertible, the inclusion $i : C ↪ X$ is a closed immersion, and there is a short exact sequence $0 \to O_{X} (- C) \to O_{X} \to i_{*} O_{C} \to 0$ with $O_{X} (- C)$ invertible (Effective cartier divisor, Invertible sheaf of cartier divisor, Effective Cartier divisors give a short exact sequence, Closed immersions of schemes).

[F2]

The curve $C$ is a proper $k$ -scheme of dimension at most one: its irreducible components are minimal primes over the principal ideals cut out by local equations of $C$ , hence have height one by the principal ideal theorem, so $\dim C \leq \dim X - 1 \leq 1$ (A minimal prime over a principal nonzerodivisor has height one, Chain dimension and the empty-space convention); $C$ is a closed subscheme of the proper $k$ -scheme $X$ , hence proper over $k$ (Proper morphisms). The degree $\deg_{C}$ is therefore defined on invertible $O_{C}$ -modules (Degree of an invertible sheaf on a proper one-dimensional scheme), and $X$ is Noetherian, locally Noetherian and of finite type over $k$ (Locally Noetherian and Noetherian schemes).

[F3]

Twisting the exact sequence of [F1] by an invertible $O_{X}$ -module $N$ gives a short exact sequence $0 \to N (- C) \to N \to i_{*} (N ∣_{C}) \to 0$ with $N (- C) = N \otimes O_{X} (- C)$ and $N ∣_{C} = i^{*} N$ (Twisting the exact sequence of an effective Cartier divisor); the Euler characteristic is additive in short exact sequences of coherent modules on the proper $k$ -scheme $X$ (Euler characteristic is additive in short exact sequences), and the closed-immersion projection formula identifies $χ (X, i_{*} F) = χ (C, F)$ for coherent $F$ on $C$ (Projection formula for a closed immersion and an invertible sheaf). Hence $χ (C, N ∣_{C}) = χ (X, N) - χ (X, N (- C))$ for every invertible $N$ .

[F4]

Definition of the values: writing $O_{X} (- C) = O_{X} (C)^{\lor}$ and $O_{X} (- D) = O_{X} (D)^{\lor}$ and $O_{X} (- C - D) = O_{X} (- C) \otimes O_{X} (- D)$ , the definition of the intersection product gives $C \cdot D = χ (X, O_{X}) - χ (X, O_{X} (- C)) - χ (X, O_{X} (- D)) + χ (X, O_{X} (- C - D)),$ and $\deg_{C} (M) = χ (C, M) - χ (C, O_{C})$ for invertible $M$ ; the defining expression is symmetric in the two divisors (Intersection numbers of Cartier divisors on a smooth projective surface). By part 2 of Degree is additive on invertible sheaves over a proper curve, $\deg_{C} (M^{\lor}) = - \deg_{C} (M)$ for invertible $M$ on the proper curve $C$ of dimension at most one.

[F5]

The smooth case: if $C$ is a smooth proper geometrically integral curve over $k$ and $M$ is an invertible $O_{C}$ -module, then $M ≅ O_{C} (D^{'})$ for a divisor $D^{'}$ on $C$ (Cartier and Weil divisors agree on a smooth curve), and $χ (C, O_{C} (D^{'})) - χ (C, O_{C}) = \deg_{k} (D^{'})$ (Riemann-Roch in Euler-characteristic form: the degree shift); hence $\deg_{C} (M)$ equals the closed-point divisor degree $\deg_{k} (D^{'})$ of Degree divisor proper curve.

[F6]

Empty case: if $C = \emptyset$ then $O_{X} (- C) = O_{X}$ and $i_{*} O_{C} = 0$ , so the defining expression for $C \cdot D$ is $χ (O_{X}) - χ (O_{X}) - χ (O_{X} (- D)) + χ (O_{X} (- D)) = 0$ (Effective cartier divisor, Effective Cartier divisors give a short exact sequence).

[F7]

The Axiom of Choice enters through the Euler-characteristic, devissage, closed-immersion and curve-degree suppliers; no selection is made in the computations below.

Proof

technique · direct; substitute the two twisting sequences into the defining four-term expression and identify the result with a degree

1.1F1F2F3

Set-up. The curve $C$ is a proper $k$ -scheme of dimension at most one, so $\deg_{C}$ is defined on invertible $O_{C}$ -modules, and the modules $O_{X} (D) ∣_{C} = i^{*} O_{X} (D)$ and $O_{X} (- C)$ , $O_{X} (- D)$ , $O_{X} (- C - D)$ appearing below are invertible, hence coherent on the locally Noetherian schemes $C$ and $X$ . In the closed-immersion exact sequence for $C$ the third term is $i_{*} O_{C}$ , and for an invertible $N$ on $X$ the twist reads $0 \to N (- C) \to N \to i_{*} (N ∣_{C}) \to 0$ .

1.2F1F3

The chi-difference identity. For every invertible $O_{X}$ -module $N$ , additivity of $χ$ on the twisted sequence of [F1] gives $χ (X, N) = χ (X, N (- C)) + χ (X, i_{*} (N ∣_{C}))$ , and the projection formula gives $χ (X, i_{*} (N ∣_{C})) = χ (C, N ∣_{C})$ ; hence $χ (C, N ∣_{C}) = χ (X, N) - χ (X, N (- C))$ .

1.3F5

The smooth comparison. If $C$ is a smooth proper geometrically integral curve, then every invertible module on $C$ is $O_{C} (D^{'})$ for a divisor $D^{'}$ , and the Euler-characteristic degree shift identifies $\deg_{C} (O_{C} (D^{'})) = χ (C, O_{C} (D^{'})) - χ (C, O_{C})$ with $\deg_{k} (D^{'})$ ; this is the asserted agreement with the closed-point divisor degree.

1.4F6

The empty case. If $C = \emptyset$ , then the ideal sheaf of $C$ is $O_{X}$ and $i_{*} O_{C} = 0$ , so the four terms of the defining expression cancel in pairs and $C \cdot D = 0$ ; on the empty curve every degree is $0$ .

2.1F4step 1.2

The main computation. Take the identity of step 1.2 for $N = O_{X}$ and for $N = O_{X} (- D)$ : $χ (X, O_{X}) - χ (X, O_{X} (- C)) = χ (C, O_{C}), χ (X, O_{X} (- D)) - χ (X, O_{X} (- C - D)) = χ (C, O_{X} (- D) ∣_{C}),$ the second because $O_{X} (- D) (- C) = O_{X} (- C - D)$ . Substituting both into the defining expression of [F4], $C \cdot D = χ (C, O_{C}) - χ (C, O_{X} (- D) ∣_{C}) .$ By [F4] applied on $C$ , $χ (C, O_{X} (- D) ∣_{C}) - χ (C, O_{C}) = \deg_{C} (O_{X} (- D) ∣_{C})$ , and by the dual-degree identity of [F4], $\deg_{C} (O_{X} (- D) ∣_{C}) = - \deg_{C} (O_{X} (D) ∣_{C})$ because $O_{X} (- D) ∣_{C} = (O_{X} (D) ∣_{C})^{\lor}$ . Therefore $C \cdot D = \deg_{C} (O_{X} (D) ∣_{C})$ , the first identity, valid for every Cartier divisor $D$ once $C$ is effective.

3.1F4step 2.1

Both divisors effective. Assume now that $D$ is effective as well. Applying step 2.1 with the roles of $C$ and $D$ interchanged gives $D \cdot C = \deg_{D} (O_{X} (C) ∣_{D})$ , and the defining expression of the intersection product is symmetric by [F4], so $C \cdot D = D \cdot C = \deg_{D} (O_{X} (C) ∣_{D})$ . Together with step 2.1 this gives the two asserted identities for effective $C$ and $D$ .

4.1F7step 1.3step 1.4step 2.1step 3.1∎

Conclusion and choice accounting. Step 2.1 proves the general identity $C \cdot D = \deg_{C} (O_{X} (D) ∣_{C})$ for effective $C$ and arbitrary Cartier $D$ ; step 3.1 adds the second identity $\deg_{D} (O_{X} (C) ∣_{D})$ when $D$ is effective; step 1.3 proves the agreement of $\deg_{C}$ with the closed-point divisor degree on a smooth proper geometrically integral curve; and step 1.4 covers the empty curve. The Axiom of Choice enters only through the suppliers listed in [F7], in particular the Euler-characteristic additivity and projection formula of [F3], the degree and dual-degree statements of [F4] and the curve-degree comparison [F5]; no selection is made in the computations.

Depends on

Used by

Dependency tree · two levels

114 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

Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry, pre-publication version 2025-10-21 (standard reference, not scraped)
The Stacks Project, Varieties, Section 33.45 (Numerical intersections) (standard reference, not scraped)