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Suppose we have a smooth Mumford's quotient $Q//PGL_k(m)$ where $Q$ is a quasi-projective variety and $k$ is an algebraically closed field of positive characteristic. Is it true that $Q$ is also smooth ?

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I am not certain precisely what you are asking, but any linearized action of $\textbf{PGL}_{m,k}$ on a variety $Q$ that has one dense open orbit will have geometric quotient $Q//\textbf{PGL}_{m,k}$ equal to a single point, which is smooth. Of course $Q$ need not be smooth, e.g., the action of $\textbf{PGL}_{3,k}$ on the parameter space $Q\subset \mathbb{P}H^0(\mathbb{P}^2,\mathcal{O}(3))$ of nodal plane cubics (which is highly singular).

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