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The Chow ring of projective space and Bezout degrees

Example

Assume the Axiom of Choice (The Axiom of Choice) inherited from the smooth-immersion and homological suppliers. Let $k$ be a field and $n \geq 0$ . In the Chow ring $A^{*} (P_{k}^{n})$ (The intersection product and Chow ring of a smooth scheme, Chow groups of projective space) put $h : = c_{1} (O_{P^{n}} (1)) \in A^{1} (P^{n})$ (Intersection with an invertible sheaf and the first Chern class, Twisting sheaf on Proj). Then:

$A^{d} (P_{k}^{n}) = Z \cdot h^{d}$ for $0 \leq d \leq n$ and $A^{d} (P_{k}^{n}) = 0$ for $d > n$ ; under the identification $A^{d} (X) = A_{n - d} (X)$ of the intersection product with the cycle groups, the class $h^{d}$ corresponds to the class $[Λ^{n - d}]$ of a linear subspace of codimension $d$ : equivalently $h^{d} \cap [P^{n}] = [Λ^{n - d}]$ . In particular $A^{*} (P_{k}^{n}) ≅ Z [h] / (h^{n + 1}),$ the isomorphism sending $h$ to $c_{1} (O (1))$ .
(Degrees of products) The degree isomorphism $\deg : A^{n} (P_{k}^{n}) \to Z$ of Chow groups of projective space satisfies $\deg (h^{n}) = 1$ . If $H_{1}, \dots, H_{n} \subseteq P^{n}$ are reduced hypersurfaces of degrees $d_{1}, \dots, d_{n}$ (degree projective hypersurface) with $H_{i} = V (f_{i})$ for nonconstant square-free forms $f_{i}$ of degree $d_{i}$ , then $[H_{i}] = d_{i} h$ in $A^{1} (P^{n})$ and $[H_{1}] \cdot [H_{2}] \dots [H_{n}] = (d_{1} d_{2} \dots d_{n}) h^{n}, \deg ([H_{1}] \dots [H_{n}]) = d_{1} d_{2} \dots d_{n} .$

Discussion. This is the Chow-ring form of Bezout's theorem: the degree of the product of the hypersurface classes is the Bezout number. The identification of this product class with the cycle of the scheme-theoretic intersection of the $H_{i}$ , with its local intersection multiplicities, is the classical proper-intersection theorem and is not claimed here; for plane curves ( $n = 2$ , curves without common components) the corresponding local-multiplicity statement is developed on the plane-curves page.

Verification

Given: the Axiom of Choice; a field $k$ ; $n \geq 0$ ; the projective space $P_{k}^{n}$ with its ample generator $O (1)$ and $h = c_{1} (O (1))$ ; reduced hypersurfaces $H_{1}, \dots, H_{n}$ of degrees $d_{1}, \dots, d_{n}$ .

[L1] The projective bundle formula gives, for a rank- $(n + 1)$ bundle $E$ on a scheme, the isomorphism $⨁_{i = 0}^{n} A_{d + i} (X) \to A_{d + n} (P (E))$ via $ξ$ -caps (The projective bundle formula for Chow groups); the projective space $P_{k}^{n}$ is the projectivization of the free rank- $(n + 1)$ bundle on $Spec k$ (Twisting sheaf on Proj, Invertible twists for degree-one generated rings).

[L2] The Chow ring structure and the identification $A^{d} = A_{n - d}$ are as in The intersection product and Chow ring of a smooth scheme; the cycle groups of projective space are $Z$ in each dimension with $\deg (h^{n}) = 1$ (Chow groups of projective space).

[L3] The cap action of $c_{1} (O (1))$ is cutting with a hyperplane $H$ : for an integral $V \subseteq̸ H$ one has $c_{1} (O (1)) \cap [V] = [V \cap H]$ (Intersection with an invertible sheaf and the first Chern class).

[L4] A reduced hypersurface $H = V (f)$ of degree $d$ has $[H] = d h$ : $c_{1} (O (1))$ is additive and normalized so that the divisor of a degree- $d$ form is $d$ times a hyperplane class (degree projective hypersurface, Chern classes of a vector bundle on a smooth scheme, Additivity, naturality and the splitting principle for Chern classes).

1.1L1L2givenalgebra

The ring. Apply [L1] to the free rank- $(n + 1)$ bundle on $Spec k$ , whose projectivization is $P_{k}^{n}$ : the formula gives $A_{d + n} (P^{n}) = ⨁_{i = 0}^{n} ξ^{i} \cap π^{*} A_{d + i} (k)$ with $ξ = c_{1} (O (1))$ , so $A^{d} (P^{n}) = Z \cdot h^{d}$ for $0 \leq d \leq n$ and $A^{d} = 0$ for $d > n$ ; the relation $h^{n + 1} = 0$ and the absence of other relations give $A^{*} (P_{k}^{n}) ≅ Z [h] / (h^{n + 1})$ with $h \mapsto c_{1} (O (1))$ .

1.2L2L3givenalgebra

Identification with linear subspaces. By induction on $d$ : for a linear subspace $Λ^{n - d + 1}$ and a general hyperplane $H$ of complementary position, $H \supseteq̸ Λ$ and $H \cap Λ = Λ^{n - d}$ is a linear subspace, so [L3] gives $c_{1} (O (1)) \cap [Λ^{n - d + 1}] = [Λ^{n - d}]$ ; starting from $h^{0} \cap [P^{n}] = [P^{n}]$ this shows $h^{d} \cap [P^{n}] = [Λ^{n - d}]$ under the identification $A^{d} = A_{n - d}$ . In particular $\deg (h^{n}) = \deg [Λ^{0}] = 1$ by [L2].

2.1L2L4step 1.2algebra∎

Degrees of products. By [L4] each reduced hypersurface of degree $d_{i}$ has class $[H_{i}] = d_{i} h$ ; multiplicativity of the Chow ring product gives $[H_{1}] \dots [H_{n}] = (d_{1} \dots d_{n}) h^{n}$ , and the degree homomorphism of [L2] sends $h^{n}$ to $1$ , so $\deg ([H_{1}] \dots [H_{n}]) = d_{1} d_{2} \dots d_{n}$ . This is the Bezout number in the Chow ring; the identification with the cycle of the scheme-theoretic intersection with local multiplicities is deliberately not asserted here.

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Sources

The Stacks Project, Chow Homology and Chern Classes, Sections 42.36 and 42.60-42.62 (standard reference, not scraped)
Ravi Vakil, Math 245 Topics in Algebraic Geometry: Introduction to Intersection Theory, Class 2 and Class 16 (standard reference, not scraped)