Variation of non-reductive geometric invariant theory

Basic geometric invariant theory

Geometric invariant theory and flips

Ever since the invention of geometric invariant theory, it has been understood that the quotient it constructs is not entirely canonical, but depends on a choice: the choice of a linearization of the group action. However, the founders of the subject never made a systematic study of this dependence. In light of its fundamental and elementary nature, this is a rather surprising gap, and this pap...

An Elementary Theorem in Geometric Invariant Theory

The purpose of this note is to prove the key theorem in a construction of the arithmetic scheme of moduli M of curves of any genus. This construction, which relies heavily on Grothendieck's whole theory of schemes, may be briefly outlined as follows: first one defines the family K of tri-canonical models of curves C of genus g, any characteristic, in P?-, as a sub-scheme of one of Grothendieck'...

Geometric Invariant Theory via Cox Rings

We consider actions of reductive groups on a varieties with finitely generated Cox ring, e.g., the classical case of a diagonal action on a product of projective spaces. Given such an action, we construct via combinatorial data in the Cox ring all maximal open subsets such that the quotient is quasiprojective or embeddable into a toric variety. As applications, we obtain an explicit description...

Geometric invariant theory and projective toric varieties

We define projective GIT quotients, and introduce toric varieties from this perspective. We illustrate the definitions by exploring the relationship between toric varieties and polyhedra. Geometric invariant theory (GIT) is a theory of quotients in the category of algebraic varieties. Let X be a projective variety with ample line bundle L, and G an algebraic group acting on X, along with a lift...