Graded linearisations

Abstract

When the action of a reductive group on a projective variety has a suitable linearisation, Mumford’s geometric invariant theory (GIT) can be used to construct and study an associated quotient variety. In this article we describe how Mumford’s GIT can be extended effectively to suitable actions of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action induces a graded linearisation in a natural way. The classical examples of moduli spaces which can be constructed using Mumford’s GIT are moduli spaces of stable curves and of (semi)stable bundles over a fixed nonsingular curve. This more general construction can be used to construct moduli spaces of unstable objects, such as unstable curves or unstable bundles (with suitable fixed discrete invariants in each case, related to their singularities or Harder–Narasimhan type). In algebraic geometry it is often useful to be able to construct quotients of algebraic varieties by linear algebraic group actions; in particular moduli spaces (or stacks) can be constructed in this way. When the linear algebraic group is reductive, and we have a suitable linearisation for its action on a projective variety, we can use Mumford’s geometric invariant theory (GIT) to construct and study such quotient varieties [32]. The aim of this article is to describe how Mumford’s GIT can be extended effectively to actions of a large family of linear algebraic groups which are not necessarily reductive, with the extra data of a graded linearisation for the action. Any linearisation in the traditional sense for a reductive group action can be regarded as a graded linearisation in a natural way. When a linear algebraic group over an algebraically closed field k of characteristic 0 is a semidirect productH = U ⋊R of its unipotent radical U and a reductive subgroupR ∼= H/U which contains a central one-parameter subgroup λ : Gm → Rwhose adjoint action on the Lie algebra of U has only strictly positive weights, we will see that any linearisation for an action of H on a projective variety X becomes graded if it is twisted by an appropriate (rational) character, and then many of the good properties of Mumford’s GIT hold. Many non-reductive linear algebraic group actions arising in algebraic geometry are actions of groups of this form: for example, any parabolic subgroup of a reductive group has this form, as does the automorphism group of any complete simplicial toric variety [11], and the group of k-jets of germs of biholomorphisms of (C, 0) for any positive integers k and p [6]. Example 0.1. The automorphism group of the weighted projective plane P(1, 1, 2) with weights 1,1 and 2 is Aut(P(1, 1, 2)) ∼= R⋉ U where R ∼= (GL(2)×Gm)/Gm ∼= GL(2) is reductive and U ∼= (k+)3 is unipotent with elements given by (x, y, z) 7→ (x, y, z + λx2 + μxy + νy2) for (λ, μ, ν) ∈ k3. Early work on this project was supported by the Engineering and Physical Sciences Research Council [grant numbers GR/T016170/1,EP/G000174/1]. Brent Doran was partially supported by Swiss National Science Foundation Award 200021-138071.