Geometric Invariant Theory (GIT) is a branch of algebraic geometry that studies the action of group actions on algebraic varieties, particularly focusing on understanding the properties of orbits and established notions of stability. It was developed primarily in the 1950s by mathematician David Mumford, building on ideas from group theory, algebraic geometry, and representation theory.
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Geometric invariant theory constructs algebraic quotients of group actions by a reductive algebraic group using invariant sections and a chosen linearization of a line bundle. An affine quotient is the spectrum of the polynomial invariant ring; a projective quotient uses homogeneous invariants and excludes the points where all positive-degree invariants vanish. Stability separates the orbits which give well-behaved geometric quotient points.