---
tag: RW85C34GQW
kind: concept
name: Group action on a variety
msc: [14L30]
specializes: [0WCY42EV9V]
sources:
  - url: https://en.wikipedia.org/wiki/Group_action
---

## Summary

An action of an [[algebraic group|GH81SYAPPN]] $G$ on a [[variety|HV6YBS0YDH]] $X$ is a [[group action|0WCY42EV9V]] given by a morphism $G \times X \to X$.
Orbits, stabilizers and fixed points of such actions are again algebraic objects.
Quotients by such actions are the subject of geometric invariant theory.
Many important varieties, such as projective spaces, Grassmannians and toric varieties, are best understood through a group action.

## Explanations

### L1 human

A group acts on a space when each group element moves the points of the space, compatibly with the group law.
In algebraic geometry one asks that this motion be given by polynomial formulas in the group element and the point together.
For example, invertible matrices act on vectors by matrix multiplication, and the formula is polynomial in the entries.
The nonzero numbers act on the plane by scaling, and the orbits are the lines through the origin with the origin removed, together with the origin itself.