Let $G=PGL(N)$ acting on a scheme $X$ over a field $k$ and $L$ be a $G$-linearized invertible sheaf. Let $X^{ss}(L)$ be the semistable locus. We know that a uniform categorical quotient $\phi:X^{ss}(L)\rightarrow X^{ss}(L)/G$ exists. Furthermore the stable locus $X^s(L)$ gives a principal bundle $\phi|_{X^s(L)}:X^s(L)\rightarrow X^s(L)/G$. By definition $\phi|_{X^s(L)}$ is flat. Is there an example where $\phi$ fails to be flat? or can one show that $\phi$ is also flat under nice hypotheses, like $X^{ss}(L)$ and $X^{ss}(L)/G$ have nice singularities?
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$\begingroup$ the orbits parametrized by the stable locus have the correct dimension, but the ones associated to the strictly semistable locus have, usually, a lower dimension. I would expect a failure of flatness on those grounds. I actually expect flatness to hold if the stable and semistable locus coincides. $\endgroup$– eventuallyCommented Sep 11, 2013 at 0:53
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