Let $G$ be a finite group and consider $R=K[V]^G$ be the invariant ring of the group G over the field K of char 0. Let $f_1,\ldots f_n$ be a set of primary invariants. Is there a nice geometric characterization of the points where the Jacobian of these polynomials vanishes? For finite reflection groups, this is the reflection hyperplane arrangement...
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The points of constant rank of $f=(f_1,\dots,f_n)$ essentially are the strata of the stratification into orbit types. See here or here and references therein. 

