Graded unipotent groups and Grosshans theory

TL;DR

This paper extends Grosshans theory to graded unipotent groups, showing that under certain conditions, invariants form finitely generated algebras and geometric quotients can be constructed, generalizing classical results.

Contribution

It introduces a framework for embedding graded unipotent groups into linear groups with Grosshans-like properties, enabling the construction of quotients and invariants.

Findings

Invariants form finitely generated algebras under certain linearizations.

Existence of projective quotients for actions of graded unipotent groups.

Construction of projective completions of geometric quotients.

Abstract

Let $U$ be a unipotent group which is graded in the sense that it has an extension $H$ by the multiplicative group of the complex numbers such that all the weights of the adjoint action on the Lie algebra of $U$ are strictly positive. We study embeddings of $H$ in a general linear group $G$ which possess Grosshans-like properties. More precisely, suppose $H$ acts on a projective variety $X$ and its action extends to an action of $G$ which is linear with respect to an ample line bundle on $X$. Then, provided that we are willing to twist the linearisation of the action of $H$ by a suitable (rational) character of $H$, we find that the $H$-invariants form a finitely generated algebra and hence define a projective variety $X/\!/H$; moreover the natural morphism from the semistable locus in $X$ to $X/\!/H$ is surjective, and semistable points in $X$ are identified in $X/\!/H$ if and only if the closures of their $H$-orbits meet in the semistable locus. A similar result applies when we replace $X$ by its product with the projective line; this gives us a projective completion of a geometric quotient of a $U$-invariant open subset of $X$ by the action of the unipotent group $U$.