---
tag: ZPZA1FKZZC
kind: concept
name: Enumerative geometry
msc: [14N10, 14N15]
sources:
  - url: https://en.wikipedia.org/wiki/Enumerative_geometry
same_as:
  wikidata: Q2429676
  wikipedia: Enumerative geometry
---

## Summary

Enumerative geometry is the branch of algebraic geometry that counts geometric objects satisfying given conditions.
Typical questions ask how many lines meet four given lines in space, or how many conics are tangent to five given conics.
Its tools include [[intersection theory|4HV4WTAX9P]], [[Schubert calculus|FWGHMADJR4]] and [[Gromov–Witten invariants|XVJA0JD79P]].
It is a meeting point of algebraic geometry, combinatorics and physics.

## Explanations

### L1 human

How many circles touch three given circles?
The answer, found by Apollonius, is $8$ in general.
Enumerative geometry asks questions of this kind: count the curves or surfaces with prescribed properties.
The answers are expected to be finite numbers that do not change when the given data move a little, as long as they stay general.
Getting them right is subtle, and some classical answers were wrong for a long time.