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Riemann-Roch for smooth projective surfaces

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $k$ be a field, let $X$ be an integral smooth projective surface over $k$ , and let $K_{X}$ be a canonical divisor (The canonical divisor of a smooth projective surface). Then for every invertible $O_{X}$ -module $L$ (Invertible sheaves) $χ (X, L) = χ (X, O_{X}) + \frac{1}{2} (L \cdot L - L \cdot K_{X}),$ and equivalently, for every Cartier divisor $D$ on $X$ (Cartier divisor) with $O_{X} (D) ≅ L$ , $χ (X, O_{X} (D)) = χ (X, O_{X}) + \frac{1}{2} D \cdot (D - K_{X}) .$ Here $χ$ is the Euler characteristic (Euler characteristic of a coherent sheaf). The right-hand side is an integer and is independent of the choice of canonical divisor and of the representative divisor of $L$ (The canonical divisor of a smooth projective surface). No ampleness, effectivity or vanishing hypothesis is imposed on $L$ .

Facts & Assumptions

Given: a field $k$ , an integral smooth projective surface $X$ over $k$ , a canonical divisor $K_{X}$ , and an invertible sheaf $L$ .

[F1]

The intersection product is defined by the alternating sum $L_{1} \cdot L_{2} = χ (X, O_{X}) - χ (X, L_{1}^{\lor}) - χ (X, L_{2}^{\lor}) + χ (X, L_{1}^{\lor} \otimes L_{2}^{\lor})$ of Intersection numbers of Cartier divisors on a smooth projective surface, is symmetric and $Z$ -bilinear, and satisfies $L_{1}^{\lor} \otimes L_{1} ≅ O_{X}$ (The surface intersection product is symmetric and bilinear, Dual of a line bundle is its tensor inverse, Tensor product of sheaves of modules, Invertible sheaves).

[F2]

The canonical sheaf satisfies $O_{X} (K_{X}) ≅ ω_{X} = ⋀^{2} Ω_{X / k}^{1}$ , and $K_{X} \cdot L = ω_{X} \cdot L$ for every invertible $L$ (The canonical divisor of a smooth projective surface).

[F3]

Serre duality: for every invertible sheaf $E$ the cup-product pairing $H^{q} (X, E) \times H^{2 - q} (X, E^{\lor} \otimes ω_{X}) \to k$ is perfect with finite-dimensional groups, so $χ (X, E) = χ (X, E^{\lor} \otimes ω_{X})$ ; in particular $χ (X, ω_{X}) = χ (X, O_{X})$ (Serre duality for locally free sheaves on a smooth projective variety, Euler characteristic of a coherent sheaf).

[F4]

The Axiom of Choice is inherited from the Serre-duality and Euler-characteristic suppliers of [F3]; the sheaf $L$ and the divisor $K_{X}$ are given data.

Proof

technique · direct: evaluate the defining alternating sum on the pair $(\mathcal L^{\vee},\mathcal L\otimes\omega_X^{\vee})$ and replace two terms by Serre duality

1.1F1

The defining sum. Put $A : = L^{\lor}$ and $B : = L \otimes ω_{X}^{\lor}$ . By [F1], $A \cdot B = χ (X, O_{X}) - χ (X, A^{\lor}) - χ (X, B^{\lor}) + χ (X, A^{\lor} \otimes B^{\lor}) = χ (X, O_{X}) - χ (X, L) - χ (X, L^{\lor} \otimes ω_{X}) + χ (X, ω_{X}),$ because $A^{\lor} ≅ L$ , $B^{\lor} ≅ L^{\lor} \otimes ω_{X}$ and $A^{\lor} \otimes B^{\lor} ≅ ω_{X}$ (Dual of a line bundle is its tensor inverse).

2.1F3step 1.1

Serre duality in the two last terms. By [F3] with $E = L^{\lor} \otimes ω_{X}$ , whose dual twisted by $ω_{X}$ is $L$ , we have $χ (X, L^{\lor} \otimes ω_{X}) = χ (X, L)$ ; and $χ (X, ω_{X}) = χ (X, O_{X})$ . Hence $A \cdot B = 2 (χ (X, O_{X}) - χ (X, L)) .$

3.1F1F2step 1.1step 2.1

Bilinearity. By [F1] and [F2], $A \cdot B = L^{\lor} \cdot (L \otimes ω_{X}^{\lor}) = - L \cdot L + L \cdot ω_{X} = L \cdot K_{X} - L \cdot L$ , where $L \cdot K_{X}$ is the intersection with $ω_{X}$ by [F2]. Comparing with step 2.1 and rearranging, $2 (χ (X, O_{X}) - χ (X, L)) = L \cdot K_{X} - L \cdot L, that is χ (X, L) = χ (X, O_{X}) + \frac{1}{2} (L \cdot L - L \cdot K_{X}) .$ The right-hand side is an integer because $χ$ is; it depends only on the isomorphism class of $L$ and on the canonical class, hence is independent of the chosen representative divisor and canonical divisor (The canonical divisor of a smooth projective surface).

4.1F4step 3.1∎

The divisor form. If $D$ is a Cartier divisor with $O_{X} (D) ≅ L$ , then $L \cdot L = D \cdot D$ and $L \cdot K_{X} = D \cdot K_{X}$ by the dictionary of Intersection numbers of Cartier divisors on a smooth projective surface and The canonical divisor of a smooth projective surface, so $χ (X, O_{X} (D)) = χ (X, O_{X}) + \frac{1}{2} D \cdot (D - K_{X})$ . The Axiom of Choice is inherited from [F4]; no further selection is made.

Depends on

Used by

Dependency tree · two levels

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Sources

Ravi Vakil, The Rising Sea: Foundations of Algebraic Geometry, pre-publication version 2025-10-21 (standard reference, not scraped)
A. Kumar and K. Venkatram, MIT 18.727 Topics in Algebraic Geometry: Algebraic Surfaces, Spring 2008, Lecture 2 (standard reference, not scraped)