---
tag: 4HV4WTAX9P
kind: concept
name: Intersection theory
msc: [14C17, 14C15]
sources:
  - url: https://en.wikipedia.org/wiki/Intersection_theory
same_as:
  wikidata: Q15176852
  wikipedia: Intersection theory
---

## Summary

Intersection theory studies how subvarieties of an algebraic variety meet, and assigns numbers and classes to their intersections.
Its central object is the Chow ring of a smooth variety, made of the [[Chow groups|EBEF6PEK4N]].
Its product sends two cycle classes to the class of their intersection, counted with [[intersection multiplicity|GPQF9GK4T9]].
It gives a rigorous form to classical enumerative geometry, including [[Bézout's theorem|KWNVF194D8]] and [[Schubert calculus|FWGHMADJR4]].
It is also the home of [[Chern classes|PAD37GKDPD]] and of the [[Hirzebruch–Riemann–Roch theorem|G3X02FKQCR]].

## Explanations

### L1 human

Two curves in the plane usually meet in finitely many points, and one wants to know how many.
The answer is stable only if one counts with multiplicity, works over the complex numbers, and includes points at infinity.
Intersection theory does this in any dimension, and it allows one to move cycles to general position without changing the answer.
On the projective plane, a curve of degree $d$ and a curve of degree $e$ have intersection number $de$.
The reason is that each is equivalent to a multiple of a line, and two lines meet in one point.