Hilbert–Mumford criterion

In mathematics, the Hilbert–Mumford criterion, introduced by David Hilbert[1] and David Mumford, characterizes the semistable and stable points of a group action on a vector space in terms of eigenvalues of 1-parameter subgroups (Dieudonné & Carrell 1970, 1971, p.58).

Let G be a reductive group acting linearly on a vector space V, a non-zero point of V is called When G is the multiplicative group

, e.g. C* in the complex setting, the action amounts to a finite dimensional representation

Then for each point x, we look at the set of weights in which it has a non-zero component.

The Hilbert–Mumford criterion essentially says that the multiplicative group case is the typical situation.

Precisely, for a general reductive group G acting linearly on a vector space V, the stability of a point x can be characterized via the study of 1-parameter subgroups of G, which are non-trivial morphisms

The standard example is the action of C* on the plane C2 defined as

Thus by the Hilbert–Mumford criterion, a non-zero point on the x-axis admits 1 as its only weight, and a non-zero point on the y-axis admits -1 as its only weight, so they are both unstable; a general point in the plane admits both 1 and -1 as weights, so it is stable.

For example, consider a set of n points on the rational curve P1 (more precisely, a length-n subscheme of P1).

The automorphism group of P1, PSL(2,C), acts on such sets (subschemes), and the Hilbert–Mumford criterion allows us to determine the stability under this action.

We can linearize the problem by identifying a set of n points with a degree-n homogeneous polynomial in two variables.

We consider therefore the action of SL(2,C) on the vector space

, we can choose coordinates x and y so that the action on P1 is given as For a homogeneous polynomial of form

So the polynomial admits both positive and negative (resp.

non-positive and non-negative) weights if and only if there are terms with i>n/2 and i

If we repeat over all the 1-parameter subgroups, we may obtain the same condition of multiplicity for all points in P1.

By the Hilbert–Mumford criterion, the polynomial (and thus the set of n points) is stable (resp.

A similar analysis using homogeneous polynomial can be carried out to determine the stability of plane cubics.

The Hilbert–Mumford criterion shows that a plane cubic is stable if and only if it is smooth; it is semi-stable if and only if it admits at worst ordinary double points as singularities; a cubic with worse singularities (e.g. a cusp) is unstable.