Reductive Affine Invariant Theory and Geometric Quotients — Examples

For $t\in {\mathbf{G}}_m$ and a monomial $x^a{y}^b$ the function action of [F1] gives $(t\cdot (x^a{y}^b))(x,y)=(t^{-1}x{)}^a(ty{)}^b=t^{b-a}{x}^a{y}^b$, so the multiplicative group acts on each monomial by the scalar $t^{b-a}$: monomials on the diagonal $a=b$ are fixed, while off-diagonal monomials are scaled by a nonconstant character.

An element $f={\sum }_{a,b}{c}_{ab}{x}^a{y}^b\in \mathbb{C}[x,y]$ is invariant under ${\mathbf{G}}_m$ if and only if ${\sum }_{a,b}{c}_{ab}(t^{b-a}-1)x^a{y}^b=0$ for every $t\in {\mathbf{G}}_m$; by step 1.1 this is the difference between $t\cdot f$ and $f$, and by uniqueness of the monomial expansion it holds if and only if $c_{ab}{t}^{b-a}=c_{ab}$ for all pairs $(a,b)$ and all $t\in {\mathbf{G}}_m$.

The coefficient conditions hold if and only if $c_{ab}=0$ for every pair with $a\mathrm{\ne }b$. If $f$ is invariant, fix a pair with $a\mathrm{\ne }b$; evaluating the condition of step 2.1 at the nonzero complex number $t=2$ gives $c_{ab}(2^{b-a}-1)=0$, and $2^{b-a}\mathrm{\ne }1$ because $2^{b-a}>1$ for $b>a$ and $0<2^{b-a}<1$ for $b<a$; hence $c_{ab}=0$. Conversely, if $c_{ab}=0$ whenever $a\mathrm{\ne }b$, then $f={\sum }_a{c}_a{x}^a{y}^a$ is a sum of diagonal monomials, and by step 1.1 each such monomial satisfies $t\cdot (x^a{y}^a)=t^{a-a}{x}^a{y}^a=x^a{y}^a$, so $t\cdot f=f$ for every $t$ and $f$ is invariant.

Consequently a polynomial $f\in \mathbb{C}[x,y]$ is ${\mathbf{G}}_m$-invariant if and only if its expansion has no off-diagonal terms, that is, if and only if $f={\sum }_a{c}_a{x}^a{y}^a$ lies in the subalgebra $\mathbb{C}[xy]$ generated by the invariant function $xy$. Hence $\mathbb{C}[x,y{]}^{{\mathbf{G}}_m}=\mathbb{C}[xy]$. The substitution $\phi :\mathbb{C}[z]\to \mathbb{C}[x,y]$, $z\mapsto xy$, is injective: if $\phi (g)=0$ for $g={\sum }_n{d}_n{z}^n$, then ${\sum }_n{d}_n{x}^n{y}^n=0$, and uniqueness of the monomial expansion forces every $d_n=0$, so $g=0$. Therefore $\mathbb{C}[xy]\cong \mathbb{C}[z]$ is a polynomial ring in one variable, and the plane's invariant ring is $\mathbb{C}[xy]$.

The trivial action of $G={\mathbf{G}}_m$ on $X=\{0\}$ is algebraic by [F2]; its unique orbit is $\{0\}$, which is closed in $X$ because every subset of the one-point space is closed, and the stabilizer of $0$ is $G_0=G$.

The group ${\mathbf{G}}_m={\mathbb{C}}^{\times }$ is infinite, since it contains $t$ for every nonzero complex number $t$; hence the stabilizer $G_0$ is not finite and the point $0$ is not stable by [F1], although its orbit is closed. This refutes the displayed assertion.

First ${\mathbf{G}}_m$ is reductive: every non-trivial subgroup $U$ has an element $u\mathrm{\ne }1$, and its rational scalar action on the one-dimensional space $\mathbb{C}$ has no non-zero fixed vector, since $(u-1)v=0$ forces $v=0$. Thus no such closed subgroup is unipotent in the fixed-vector sense of [F6]. Part (i): by [F1] the invariant ring is $\mathbb{C}[xy]$, a polynomial ring in the single variable $xy$, and by [F2] the quotient is $\pi :X\to X\mathrm{/}\text{\hspace{0.17em}⁣}\mathrm{/}G=\mathbb{C}$ with $\pi (x,y)=xy$, a categorical quotient.

Part (ii): for $c\mathrm{\ne }0$ the fibre ${\pi }^{-1}(c)=\{(x,y):xy=c\}$ is a single orbit, because for $(x,y),(x^{\mathrm{\prime }},y^{\mathrm{\prime }})$ with $xy=x^{\mathrm{\prime }}{y}^{\mathrm{\prime }}=c$ the element $t=x^{\mathrm{\prime }}\mathrm{/}x$ is nonzero and $t\cdot (x,y)=(x^{\mathrm{\prime }},t^{-1}y)=(x^{\mathrm{\prime }},(x\mathrm{/}{x}^{\mathrm{\prime }})y)=(x^{\mathrm{\prime }},c\mathrm{/}{x}^{\mathrm{\prime }})=(x^{\mathrm{\prime }},y^{\mathrm{\prime }})$. The hyperbola $xy=c$ is closed in the plane, being the zero set of the polynomial $xy-c$. On $\{x=0,y\mathrm{\ne }0\}$ the action is transitive by the second coordinate and the orbit is not closed: its closure contains the origin, and a polynomial vanishing on $\{x=0,y\mathrm{\ne }0\}$ vanishes on the whole axis $\{x=0\}$ because a one-variable polynomial with infinitely many roots is zero; similarly for $\{y=0,x\mathrm{\ne }0\}$. The origin is fixed by the action, so it is an orbit, it is closed, and it is the unique closed orbit in the zero fibre by [F2]; the zero fibre consists of exactly the two punctured axes and the origin.

Stabilizers and stability: at a point with $x\mathrm{\ne }0$ the condition $tx=x$ gives $t=1$, and at a point with $y\mathrm{\ne }0$ the condition $t^{-1}y=y$ gives $t=1$; hence every point of $X\setminus \{0\}$ has trivial stabilizer, while the origin has stabilizer ${\mathbf{G}}_m={\mathbb{C}}^{\times }$, which is infinite. By [F4] and step 1.2 the stable points are exactly those with $xy\mathrm{\ne }0$: there the orbit is a closed hyperbola and the stabilizer is trivial, while the punctured-axis points and the origin are unstable.

Part (iii): by step 2.1 the stable locus is $X^s=\{xy\mathrm{\ne }0\}$ and $\pi (X^s)={\mathbb{C}}^{\times }$, and by [F3] the restriction ${\pi }^s:X^s\to {\mathbb{C}}^{\times }$ is a geometric quotient whose fibres are exactly the orbits and whose orbits are closed in $X$.

The bundle statement: the map $\mathrm{\Phi }:{\mathbf{G}}_m\times {\mathbf{G}}_m\to X^s$, $\mathrm{\Phi }(t,c)=(t,c\mathrm{/}t)$, is an isomorphism with inverse $(x,y)\mapsto (x,xy)$, and it is $G$-equivariant for the action on the first factor because $\mathrm{\Phi }(t^{\mathrm{\prime }}t,c)=(t^{\mathrm{\prime }}t,c\mathrm{/}(t^{\mathrm{\prime }}t))=t^{\mathrm{\prime }}\cdot (t,c\mathrm{/}t)=t^{\mathrm{\prime }}\cdot \mathrm{\Phi }(t,c)$. Under $\mathrm{\Phi }$ the quotient ${\pi }^s$ corresponds to the projection ${\mathbf{G}}_m\times {\mathbf{G}}_m\to {\mathbf{G}}_m$, $(t,c)\mapsto c$, which is a trivial principal ${\mathbf{G}}_m$-bundle with structure group acting on the first factor and a geometric quotient; so ${\pi }^s$ is a principal ${\mathbf{G}}_m$-bundle in this explicit trivialized sense, and no identification with any other principal-bundle theory is used.

Assembly: (i) is step 1.1, (ii) is step 1.2 together with the closed-orbit count of [F2], (iii) is steps 2.1, 3.1 and 4.1; the origin is unstable although its orbit is closed because its stabilizer is all of ${\mathbf{G}}_m$. All Axiom of Choice content is inherited from the affine quotient and stable-locus suppliers.