Introduction

Geometric Invariant Theory is a useful technique to construct quotients by group actions in Algebraic Geometry. It is widely applied in many branches of Mathematics, in particular in the theory of moduli spaces.

For example, suppose you want to classify smooth curves of fixed genus ${\displaystyle g\geq 2}$ up to isomorphism; then you can embed any one of them in projective space (with the same numerical invariants!), so you can regard them as (some; and understanding which is an important part of the theory) points of a Hilbert scheme (this is the scheme representing closed subschemes of ${\displaystyle \mathbb {P} ^{N}}$ with a fixed Hilbert polynomial; the easiest examples are ${\displaystyle \mathbb {P} ^{m}}$ identifying hypersurfaces with their coefficients and ${\displaystyle \operatorname {Gr} (k,n)}$ parametrizing linear ${\displaystyle k}$-dimensional subspaces of ${\displaystyle \mathbb {C} ^{n}}$). But you did not care about the polarization to start with, therefore you must now take the quotient by the action of ${\displaystyle PGL_{N}}$ changing coordinates of ${\displaystyle \mathbb {P} ^{N}}$.

For simplicity we will work over ${\displaystyle \mathbb {C} }$.