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).

Definition of stability

When the weight on the fibre over the limit x₀ is positive, the point x is taken to 0 along the C^* action and the orbit closure contains 0. When the weight is positive, x goes off to infinity, and the orbit is closed.

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

semi-stable if 0 is not contained in the closure of its orbit, and unstable otherwise;
stable if its orbit is closed, and its stabilizer is finite. A stable point is a fortiori semi-stable. A semi-stable but not stable point is called strictly semi-stable.

When G is the multiplicative group $\mathbb {G} _{m}$ , e.g. C^* in the complex setting, the action amounts to a finite dimensional representation $\lambda \colon \mathbf {C} ^{*}\to \mathrm {GL} (V)$ . We can decompose V into a direct sum $V=\textstyle \bigoplus _{i}V_{i}$ , where on each component V_i the action is given as $\lambda (t)\cdot v=t^{i}v$ . The integer i is called the weight. Then for each point x, we look at the set of weights in which it has a non-zero component.

If all the weights are strictly positive, then $\lim _{t\to 0}\lambda (t)\cdot x=0$ , so 0 is in the closure of the orbit of x, i.e. x is unstable;
If all the weights are non-negative, with 0 being a weight, then either 0 is the only weight, in which case x is stabilized by C^*; or there are some positive weights beside 0, then the limit $\lim _{t\to 0}\lambda (t)\cdot x$ is equal to the weight-0 component of x, which is not in the orbit of x. So the two cases correspond exactly to the respective failure of the two conditions in the definition of a stable point, i.e. we have shown that x is strictly semi-stable.

Statement

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 $\lambda \colon \mathbb {G} _{m}\to G$ . Notice that the weights for the inverse $\lambda ^{-1}$ are precisely minus those of $\lambda$ , so the statements can be made symmetric.

A point x is unstable if and only if there is a 1-parameter subgroup of G for which x admits only positive weights or only negative weights; equivalently, x is semi-stable if and only if there is no such 1-parameter subgroup, i.e. for every 1-parameter subgroup there are both non-positive and non-negative weights;
A point x is strictly semi-stable if and only if there is a 1-parameter subgroup of G for which x admits 0 as a weight, with all the weights being non-negative (or non-positive);
A point x is stable if and only if there is no 1-parameter subgroup of G for which x admits only non-negative weights or only non-positive weights, i.e. for every 1-parameter subgroup there are both positive and negative weights.

Examples and applications

The action of C^* on the plane C², with orbits being plane conics (hyperbolas).

Action of C^* on the plane

The standard example is the action of C^* on the plane C² defined as $t\cdot (x,y)=(tx,t^{-1}y)$ . Clearly the weight in the x-direction is 1 and the weight in the y-direction is -1. 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.

Points in P¹

Many examples arise in moduli problems. 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 $H^{0}({\mathcal {O}}_{\mathbf {P} ^{1}}(n))$ of such homogeneous polynomials. Given a 1-parameter subgroup $\lambda \colon \mathbf {C} ^{*}\to \mathrm {SL} (2,\mathbf {C} )$ , we can choose coordinates x and y so that the action on P¹ is given as

\lambda (t)\cdot [x:y]=[t^{k}x:t^{-k}y].

For a homogeneous polynomial of form $\textstyle \sum _{i=0}^{n}a_{i}x^{i}y^{n-i}$ , the term $x^{i}y^{n-i}$ has weight k(2i-n). 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<n/2 (resp. i≥n/2 and i≤n/2). In particular the multiplicity of x or y should be <n/2 (reps. ≤n/2). 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. semi-stable) if and only if its multiplicity at any point is <n/2 (resp. ≤n/2).

Plane cubics

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.

References

Hilbert, D. (1893), "Über die vollen Invariantensysteme (On Full Invariant Systems)", Math. Annalen, 42 (3): 313, doi:10.1007/BF01444162

Dieudonné, Jean A.; Carrell, James B. (1970), "Invariant theory, old and new", Advances in Mathematics, 4: 1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525
Dieudonné, Jean A.; Carrell, James B. (1971), Invariant Theory, Old and New, Boston, MA: Academic Press, ISBN 978-0-12-215540-6, MR 0279102
Harris, Joe; Morrison, Ian (1998), Moduli of Curves, Springer, doi:10.1007/b98867
Thomas, Richard P. (2006), "Notes on GIT and symplectic reduction for bundles and varieties", Surveys in Differential Geometry, 10, arXiv:math/0512411v3

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[1] Hilbert, D. (1893), "Über die vollen Invariantensysteme (On Full Invariant Systems)", Math. Annalen, 42 (3): 313, doi:10.1007/BF01444162