---
tag: W94AEJV9NK
kind: definition
name: Enumerative geometry
msc: [14N10, 14N15]
defines: ZPZA1FKZZC
uses: [ZT9A6JK8QF, 4HV4WTAX9P, FJZRW9P0K3]
sources:
  - url: https://en.wikipedia.org/wiki/Enumerative_geometry
---

## Definition

*Enumerative geometry* is the branch of algebraic geometry that determines the number of geometric objects of a given kind that satisfy given conditions.
The objects are typically subvarieties of [[projective space|ZT9A6JK8QF]], such as lines, conics or rational curves.
The conditions are chosen in general position, and an answer is a number that is the same for all general choices.
A standard method treats the objects as points of a [[moduli space|FJZRW9P0K3]] and each condition as a subvariety.
The count is then an [[intersection number|4HV4WTAX9P]].

## Remarks

Example: in $\mathbb{P}^3$, exactly $2$ lines meet four lines in general position.
In the Grassmannian of lines, each condition is a hypersurface, and the product of four such classes is $2$ times the class of a point.

Example: there are $3264$ smooth conics in $\mathbb{P}^2$ tangent to five general conics.
Steiner gave the answer $7776$, which counts degenerate double lines as well, and the correct number $3264$ is due to Chasles.

Non-example: the number of lines meeting four lines that all lie in one plane is infinite.
Every line in that plane meets all four, so the conditions are not in general position.

Other classical numbers are $27$ lines on a smooth cubic surface and $2875$ lines on a general quintic threefold.