Module schemes in invariant theory/Fibers and ramification/Section

We are now interested in the fibers and ramification properties of ${}Z_{\rho }\rightarrow X$.

Theorem

Let ${}G$ be a finite nonmodular group acting on a ${}K$-algebra ${}S$ with quotient scheme ${}X$. Let ${}P\in X$ be a closed point, and let

${}S/{\mathfrak {m}}_{P}\cong S_{1}\times \cdots \times S_{r}\,$

be the decomposition of the fiberring above ${}P$ into local isomorphic rings, corresponding to points ${}Q=Q_{1},\ldots ,Q_{r}$ of ${}\operatorname {Spec} {\left(S\right)}$. Let ${}H$ denote the stabilizer group of ${}Q$. Let ${}\rho$ denote a linear representation of ${}G$, and let ${}{\tilde {\rho }}$ denote the induced representation on ${}H$. Then the fiber ring of ${}Z_{\rho }$ over ${}P$ is ${}{\left(S_{1}[W_{1},\ldots ,W_{m}]\right)}^{\tilde {\rho }}$.

Corollary

If ${}G$ is nonmodular and ${}K$ is algebraically closed, then the fibers of ${}Z_{\rho }$ are homeomorphic to invariant rings over ${}K$.

The extreme cases are

${}r={\#\left(G\right)}\,.$

Then ${}H$ is trivial, and the fiber ring is ${}K[W_{1},\ldots ,W_{m}]$. This implies also that ${}Z_{\rho }\rightarrow X$ is a vector bundle over the image of the free locus of ${}G$.

${}r=1\,.$

Then ${}H=G$, and the fiber ring is ${}{\left(S/{\mathfrak {m}}_{P}S[W_{1},\ldots ,W_{m}]\right)}^{\rho }$.

Corollary

Suppose the situation of fact, and assume that ${}K$ is algebraically closed. Then the fiber of ${}Z_{\rho }$ over a point ${}P\in X$ is reduced if and only if for one (any) point ${}Q$ above ${}P$, the restriction of ${}\rho$ to the stabilizer of ${}Q$

(with respect to the basic action of ${}G$) is trivial.