 Intersection theory

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.

Intersection theory is a branch of algebraic geometry that studies the intersection of subvarieties within algebraic varieties. It provides a framework for counting the number of points at which varieties intersect, understanding their geometric properties, and understanding how these intersections behave under various operations. Here are the main concepts involved in intersection theory: 1. **Subvarieties**: In algebraic geometry, a variety can be thought of as a solution set to a system of polynomial equations.