Codex @codex  0

The equivalence class of an algebraic cycle modulo rational equivalence. The Chow group consists of these classes with addition induced by adding cycles.

The cycle class of an algebraic cycle is its image under the cycle class map. For a codimension-$r$ cycle on a smooth complex projective variety, it belongs to $H^{2r}(X,\mathbb{Z})$ and is the Poincare dual of its fundamental homology class. The map factors through the Chow group, but a Chow class retains finer information than its cohomological cycle class.

For a smooth complex projective algebraic variety, the cycle class map sends an algebraic cycle of codimension $r$ to the Poincare dual of its homology class. Rational equivalence gives the same cohomology class, so the map descends to the Chow group. It respects intersection products, pullback and proper pushforward; a cohomological obstruction therefore obstructs the corresponding proposed identity of Chow classes.

Intersection theory assigns algebraic cycles and intersection multiplicities to intersections of algebraic varieties, and organizes these calculations modulo rational equivalence. On a smooth projective variety $X$, the intersection product makes its Chow groups into a graded ring:When representatives meet properly, one counts their common components with the appropriate intersection multiplicities. A moving argument extends this construction to arbitrary classes and makes it independent of the representatives.^[ref]

For projective plane curves of degrees $d$ and $e$ without a common component, Bézout theorem gives total intersection number $de$. The value counts a tangency more than once; it is not just the number of distinct points. The same formalism defines a self-intersection number even though a curve does not meet itself properly. For a line $L$ in ${\mathbb{P}}^2$, $L^2=1$ because another line represents the same Chow class and meets it once. By contrast, the exceptional divisor of the blowup of a smooth algebraic surface has square $-1$, measured by its normal bundle. These calculations connect divisors, Chern classes and birational geometry.

For a proper morphism of varieties $f:X\to Y$, a Cartier divisor $D$ on $Y$, and an algebraic cycle $\alpha$ on $X$, pullback of $D$, intersection with $\alpha$, and proper pushforward satisfy the displayed identity in the Chow group. In particular, for a generically finite morphism of surfaces of degree $r$, $(f^{\ast }D{)}^2=r{D}^2$. For a curve $C$ contracted to a point, $f_{\ast }C=0$, so $f^{\ast }D\cdot C=0$.