---
tag: 9EG3A21ET1
kind: concept
name: Geometric invariant theory
msc: [14L24, 14L30, 14D20]
sources:
  - url: https://en.wikipedia.org/wiki/Geometric_invariant_theory
same_as:
  wikidata: Q5535491
  wikipedia: Geometric invariant theory
---

## Summary

Geometric invariant theory, or GIT, is a method for building quotients of algebraic varieties by actions of [[reductive groups|2XS84PWE3S]].
It replaces the orbit space, which is often badly behaved, by the spectrum or the Proj of a [[ring of invariants|6VTR5EBNYC]].
To get a good quotient one throws away the unstable points and keeps the semistable ones.
David Mumford developed the theory to construct [[moduli spaces|FJZRW9P0K3]], such as the moduli of curves and of vector bundles.

## Explanations

### L1 human

Suppose a group moves the points of a space around and you want a space whose points are the orbits.
Some orbits are too small or too close to others, and the naive orbit space is then not even Hausdorff.
GIT uses invariant polynomials to decide which orbits to keep: two points are identified when no invariant can separate them.
For example, the group of nonzero scalars acts on $\C^{n+1}$ by scaling.
With a suitable linearization, the origin is the only unstable point, and removing it and taking orbits gives the projective space $\mathbb{P}^n$.