5.3 Chow rings and intersection theory

Chow rings are a powerful tool in algebraic geometry, combining algebra and geometry to study varieties. They capture information about algebraic cycles and their intersections, providing a framework for understanding the structure of algebraic varieties.

Intersection theory, built on Chow rings, allows us to compute intersection numbers and solve enumerative problems. This connects to the broader theme of cohomology by providing a concrete way to study algebraic cycles and their relationships on varieties.

Chow Ring of Algebraic Varieties

Definition and Basic Properties

• The Chow ring $A(X)$ of an algebraic variety $X$ is a graded ring whose elements are equivalence classes of algebraic cycles on $X$, modulo rational equivalence
• The grading on $A(X)$ is given by the codimension of cycles: $A(X) = \oplus_i A^i(X)$, where $A^i(X)$ consists of codimension $i$ cycles modulo rational equivalence
• The product structure on $A(X)$ is given by the intersection product of cycles, which makes $A(X)$ into a commutative, associative ring with unit
• The Chow ring is an important invariant of an algebraic variety that captures information about its algebraic cycles and their intersections

Functoriality

• The Chow ring is functorial: given a morphism $f: X \to Y$ of algebraic varieties, there are induced morphisms $f^*: A(Y) \to A(X)$ (pullback) and $f_*: A(X) \to A(Y)$ (pushforward) that are ring homomorphisms
• The pullback $f^*$ and pushforward $f_*$ satisfy various compatibility conditions, such as the projection formula: $f_*(f^*(\alpha) \cdot \beta) = \alpha \cdot f_*(\beta)$

Intersection Products in the Chow Ring

Definition and Properties

• The intersection product of two cycles $\alpha \in A^i(X)$ and $\beta \in A^j(X)$ is a cycle $\alpha \cdot \beta \in A^{i+j}(X)$, defined geometrically by intersecting representatives of $\alpha$ and $\beta$ transversely and taking the resulting cycle
• The intersection product is commutative: $\alpha \cdot \beta = \beta \cdot \alpha$
• The intersection product is associative: $(\alpha \cdot \beta) \cdot \gamma = \alpha \cdot (\beta \cdot \gamma)$
• The intersection product of a cycle with the fundamental class $[X] \in A^0(X)$ is the cycle itself: $\alpha \cdot [X] = \alpha$

Computing Intersection Numbers

• The intersection product can be used to compute intersection numbers, which count the number of points (with multiplicity) in the intersection of cycles representing subvarieties
• For example, the intersection number of two curves $C$ and $D$ on a surface $S$ can be computed as the degree of the zero-cycle $[C] \cdot [D] \in A^2(S)$

Existence and Uniqueness of Intersection Product

Existence via Moving Lemma

• The existence of the intersection product can be proved using the moving lemma, which states that given cycles $\alpha$ and $\beta$ on a smooth variety $X$, there exist rationally equivalent cycles $\alpha'$ and $\beta'$ such that $\alpha'$ and $\beta'$ intersect properly (i.e., in the expected dimension)
• The moving lemma allows the construction of the intersection product by taking representatives that intersect properly and showing that the result is independent of the choice of representatives up to rational equivalence

Uniqueness via Axioms

• The uniqueness of the intersection product follows from the fact that it satisfies certain axioms, such as commutativity, associativity, and compatibility with pushforwards and pullbacks
• The intersection product can be shown to be the unique operation on the Chow ring satisfying these axioms, using the moving lemma and the properties of rational equivalence

Intersection Theory for Enumerative Geometry

Enumerative Problems and Chow Ring

• Enumerative geometry deals with counting the number of geometric objects (curves, surfaces) satisfying certain conditions, such as passing through given points or tangent to given subvarieties
• Intersection theory provides a powerful tool for solving enumerative geometry problems by translating them into computations in the Chow ring
• The basic strategy is to represent the geometric conditions as cycles in the Chow ring and compute their intersection product, which gives the number of objects satisfying the conditions (counted with multiplicity)

Classical Examples

• Bézout's theorem: the number of intersection points of two plane curves of degrees $d$ and $e$ is $de$ (counted with multiplicity)
• Schubert calculus: counting the number of linear subspaces of a given dimension satisfying certain incidence conditions with respect to fixed flags (complete flags, partial flags)

Advanced Applications

• More advanced applications of intersection theory to enumerative geometry involve the use of moduli spaces (space of curves, space of stable maps), virtual fundamental classes, and quantum cohomology
• These techniques allow the solution of more sophisticated enumerative problems, such as counting curves on threefolds or computing Gromov-Witten invariants