Linearizations of tensor powers and the equivariant section ring

Tensor powers. By [F1] the linearization is an isomorphism $\phi :{\sigma }^{\ast }L\to {\mathrm{p}\mathrm{r}}_2^{\ast }L$ whose two pullbacks to $G\times G\times X$ satisfy the cocycle identity. Taking $n$-th tensor powers and using [F3] gives an isomorphism ${\phi }^{\otimes n}:{\sigma }^{\ast }(L^{\otimes n})\to {\mathrm{p}\mathrm{r}}_2^{\ast }(L^{\otimes n})$ satisfying the same cocycle identity, and the corresponding fibre maps are $\mathbb{C}$-linear isomorphisms of lines; hence $L^{\otimes n}$ carries an induced $G$-linearization $m^{\otimes n}$, and a $G$-equivariant isomorphism $L\to L^{\mathrm{\prime }}$ of linearized invertible sheaves induces $G$-equivariant isomorphisms $L^{\otimes n}\to L^{\mathrm{\prime }\otimes n}$, which is functoriality. The canonical multiplication $L^{\otimes r}\otimes L^{\otimes s}\to L^{\otimes (r+s)}$ is the associativity identification of the same tensor power constructed in two ways, so it intertwines $m^{\otimes r}\otimes m^{\otimes s}$ with $m^{\otimes (r+s)}$: on a fibre at $x$ both sides send $(g{\mathrm{\ell }}_1,g{\mathrm{\ell }}_2)$ to $g({\mathrm{\ell }}_1{\mathrm{\ell }}_2)$.

The action on sections. For $\sigma \in \mathrm{\Gamma }(X,L^{\otimes n})$ define $(g\cdot \sigma )(x):=m^{\otimes n}(g,\sigma (g^{-1}x))$, an element of $(L^{\otimes n}{)}_{g{g}^{-1}x}=(L^{\otimes n}{)}_x$. The assignment $(g,x)\mapsto (g\cdot \sigma )(x)$ is the composition of the morphisms $(g,x)\mapsto (g,g^{-1}x)$, $\mathrm{i}\mathrm{d}\times \sigma$, and $m^{\otimes n}$, so it is a regular section of ${\mathrm{p}\mathrm{r}}_2^{\ast }(L^{\otimes n})$ over $G\times X$; in particular $g\cdot \sigma$ is a global section for each $g$. The action is linear in $\sigma$, satisfies $e\cdot \sigma =\sigma$, and $g\cdot (h\cdot \sigma )=(gh)\cdot \sigma$ by the group law of the action on the total space; restricting to a $G$-stable open subset $U\subseteq X$ commutes with the formula, so the restriction map is $G$-equivariant.

Finite dimensionality of orbits. Fix $\sigma \in \mathrm{\Gamma }(X,L^{\otimes n})$ and choose, using [F6], a finite affine cover $X=U_1\cup \cdots \cup U_{\mathrm{\ell }}$ such that $L^{\otimes n}{\mathrm{\mid }}_{U_j}$ is trivial, with trivializations ${\tau }_j$. On the affine product $G\times U_j$ the section $(g,x)\mapsto g\cdot \sigma (g^{-1}x)$ corresponds under ${\tau }_j$ to a regular function, hence by [F5] to a finite sum ${\sum }_r{h}_{j,r}(g){\phi }_{j,r}(x)$ with $h_{j,r}\in \mathbb{C}[G]$ and ${\phi }_{j,r}\in \mathcal{O}(U_j)$; write $s_{j,r}\in \mathrm{\Gamma }(U_j,L^{\otimes n})$ for the local section corresponding to ${\phi }_{j,r}$ and $V_j={\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{n}}_{\mathbb{C}}\{s_{j,r}{\}}_r$, a finite-dimensional subspace. For every $g\in G$ the restricted section $(g\cdot \sigma ){\mathrm{\mid }}_{U_j}$ lies in $V_j$, so the orbit $G\cdot \sigma$ is contained in the subspace $W_{\sigma }=\{s\in \mathrm{\Gamma }(X,L^{\otimes n}):s{\mathrm{\mid }}_{U_j}\in V_j\text{ for all }j\}$, which is finite-dimensional because restriction $\mathrm{\Gamma }(X,L^{\otimes n})\to {⨁}_j\mathrm{\Gamma }(U_j,L^{\otimes n})$ is injective by [F6].

Nonvanishing loci. For $\sigma \in \mathrm{\Gamma }(X,L^{\otimes r})$ and $\tau \in \mathrm{\Gamma }(X,L^{\otimes s})$ the identification $\sigma \otimes \tau \in \mathrm{\Gamma }(X,L^{\otimes r}\otimes L^{\otimes s})$ with its image $\sigma \tau \in \mathrm{\Gamma }(X,L^{\otimes (r+s)})$ is the canonical one, so [F4] gives $X_{\sigma \otimes \tau }=X_{\sigma }\cap X_{\tau }$. For a $G$-invariant $\sigma$ and $k\ge 1$, write ${\sigma }^{\otimes k}$ for the $k$-fold product inside $\mathrm{\Gamma }(X,L^{\otimes kr})$; trivializing $L$ near a point $x$, the section $\sigma$ corresponds to a regular function $f$ and ${\sigma }^{\otimes k}$ to $f^k$, so $f^k$ is nonzero at $x$ exactly when $f$ is; hence $X_{{\sigma }^{\otimes k}}=X_{\sigma }$.

Rationality. Let $V_{\sigma }\subseteq W_{\sigma }$ be the span of the orbit $G\cdot \sigma$; it is $G$-stable by step 1.2 and finite-dimensional by step 1.3. The orbit map $G\to V_{\sigma }$, $g\mapsto g\cdot \sigma$, is a morphism: after choosing a frame of the line fibre at $x$, each evaluation $g\mapsto (g\cdot \sigma )(x)$ is a regular scalar function by step 1.2. These scalar evaluation functionals span $V_{\sigma }^{\ast }$: a section annihilated by all of them is zero, since in a local frame its coefficient is a regular function on a reduced classical variety vanishing at every point. Hence for every ${\sigma }^{\mathrm{\prime }}\in V_{\sigma }$ the map $g\mapsto g\cdot {\sigma }^{\mathrm{\prime }}$ is a morphism, being a linear combination of orbit maps of spanning elements, and choosing a basis of $V_{\sigma }$ exhibits the action of $G$ on $V_{\sigma }$ through matrices with regular entries; thus $V_{\sigma }$ is a finite-dimensional rational $G$-module on which $G$ acts by an algebraic action, containing $\sigma$. As $\sigma$ was arbitrary, $\mathrm{\Gamma }(X,L^{\otimes n})$ is a rational $G$-module. For $n=0$ the formula reads $(g\cdot f)(x)=f(g^{-1}x)$, so constant functions are fixed; if $X$ is projective and irreducible, [F7] gives $\mathrm{\Gamma }(X,{\mathcal{O}}_X)=\mathbb{C}$ under its stated AC assumption.

Graded algebra. Define the product of homogeneous elements $\sigma \in \mathrm{\Gamma }(X,L^{\otimes r})$, $\tau \in \mathrm{\Gamma }(X,L^{\otimes s})$ by $\sigma \tau \in \mathrm{\Gamma }(X,L^{\otimes (r+s)})$ obtained from $\sigma \otimes \tau$ under the canonical isomorphism of [F3], extended bilinearly. This makes $R(X,L)$ a commutative graded $\mathbb{C}$-algebra with unit $1\in \mathrm{\Gamma }(X,{\mathcal{O}}_X)$ and degree-zero part $\mathrm{\Gamma }(X,{\mathcal{O}}_X)$, because the multiplication maps are the canonical associativity isomorphisms of tensor powers and are $\mathbb{C}$-bilinear and compatible with restriction. By step 1.1 the multiplication is $G$-equivariant, so each $g$ acts by a graded algebra automorphism, and each graded piece is a rational $G$-module by step 2.1: $R(X,L)$ is a graded rational $G$-algebra. If $X$ is projective and irreducible the degree-zero part is $\mathbb{C}$ by step 2.1.

Finally (i)-(iv) have been established: (i) in step 1.1, (ii) in steps 1.2, 1.3 and 2.1, (iii) in step 3.1, and (iv) in step 1.4. In particular every space of sections of a tensor power of a linearized invertible sheaf is a rational $G$-module and $R(X,L)$ is a graded rational $G$-algebra, as asserted.