Hilbert–Mumford criterion

In mathematics, the Hilbert–Mumford criterion, introduced by David Hilbert 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 .

Definition of stability

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 $\mathbb_m$, e.g. C^* in the complex setting, the action amounts to a finite dimensional representation $\lambda\colon \mathbf^*\to\mathrm(V)$. 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 $\lambda(t)\cdot v=t^iv$. 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 $\lim_\lambda(t)\cdot x=0$, 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 $\lim_\lambda(t)\cdot x$ 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 $\lambda\colon\mathbb_m\to G$. Notice that the weights for the inverse $\lambda^$ 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

Action of C^* on the plane

The standard example is the action of C^* on the plane C² defined as $t\cdot(x,y)=(tx,t^y)$. 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 $H^0(\mathcal_(n))$ of such homogeneous polynomials. Given a 1-parameter subgroup $\lambda\colon\mathbf^*\to \mathrm(2,\mathbf)$, we can choose coordinates ''x'' and ''y'' so that the action on P¹ is given as
:\lambda(t)\cdot :y ^kx:t^y
For a homogeneous polynomial of form $\textstyle\sum_^na_ix^iy^$, the term $x^iy^$ has weight ''k''(2''i''-''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.

See also

*Geometric invariant theory
*GIT quotient

References

*
*
*
*

{{DEFAULTSORT:Hilbert-Mumford criterion
Invariant theory