Module schemes/Singularities/Fiber/Fact

Let ${}G$ be a nonmodular group acting linearly on ${}{{\mathbb {A} }_{K}^{d}}$, and let a linear representation ${}\rho$ of ${}G$ be given with quotient scheme

$Z_{\rho }\longrightarrow X.$

Let ${}P\in X$ be a point with a preimage point ${}Q\in {{\mathbb {A} }_{K}^{d}}$.

Then the following are equivalent.

1. Every point of the fiber of ${}Z_{\rho }$ over ${}P$ is singular.
2. ${}\operatorname {Stab} (Q)\cap \operatorname {ker} {\left(\rho \right)}$ is not a reflection group via ${}\beta$.

Write a proof