Geometric invariant theory

Geometric invariant theory constructs algebraic quotients of group actions by a reductive algebraic group using invariant sections and a chosen linearization of a line bundle. An affine quotient is the spectrum of the polynomial invariant ring; a projective quotient uses homogeneous invariants and excludes the points where all positive-degree invariants vanish. Stability separates the orbits which give well-behaved geometric quotient points.

For a complex reductive algebraic group acting linearly on $V$, decompose a nonzero vector into weights for each algebraic one-parameter subgroup. With the displayed sign convention, semistability is equivalent to $\mu \ge 0$ for every such subgroup, and stability is equivalent to strict positivity for every nontrivial subgroup. One must test conjugates of the subgroups as well as those diagonal in a single fixed coordinate system.

Under $\mathrm{diag}(t,t^{-1})$, acting by inverse substitution, $X^{d-j}{Y}^j$ has weight $2j-d$. If $r$ is the smallest active exponent of $Y$, the multiplicity at $[1:0]$ is $r$ and the Hilbert-Mumford value is $d-2r$. Conjugating replaces this point by any projective point. Thus strict positivity is exactly the displayed condition; replacing the strict inequality by $m_p\le d/2$ gives semistability. A binary quartic is stable precisely when it has no repeated projective root.

A point is stable when a positive-degree homogeneous invariant is nonzero there and its orbit is closed in that invariant's nonvanishing affine open set, with finite stabilizer. This is stronger than merely having a closed orbit or a nonzero invariant. For binary forms of degree four, the stable points have four distinct projective roots.

The affine null cone is the set of vectors whose orbit closure contains the origin. For a reductive algebraic group it is also the common zero locus of all positive-degree homogeneous invariants. A nonzero vector in this set represents an unstable projective point. The Hilbert-Mumford criterion for the affine null cone detects membership by a algebraic one-parameter subgroup with only positive active weights.

For conjugation of $n$ by $n$ complex matrices by the special linear group, the affine null cone consists exactly of the nilpotent matrices. If a conjugate tends to zero along an algebraic one-parameter subgroup, its constant characteristic polynomial must be $T^n$. Conversely, put a nilpotent matrix in strictly upper triangular form and conjugate it by $\mathrm{diag}(t^{n-1},t^{n-3},\dots ,t^{1-n})$. Every active matrix entry has positive weight and tends to zero. The Hilbert-Mumford criterion for projective stability therefore identifies the semistable locus of the projective conjugation representation as the complement of the projectivized nilpotent cone.

For the natural linearization of a line bundle of the action on lines in $V$, a point is semistable when some positive-degree homogeneous invariant on $V$ is nonzero on a representative. This condition is independent of the representative. The nonsemistable vectors form the affine null cone.

A linearization of a line bundle is a group action on its total space lifting the action on the base, linear on each fibre and obeying the group law. It induces actions on global sections of its tensor powers. For the projective space of lines in a linear representation $V$, the induced action on the dual of the tautological line bundle gives the natural linearization of $\mathcal{O}(1)$. Geometric invariant theory uses its invariant sections to define the semistable locus.