This question is related to this one. If I have an locally closed, quasi projective scheme $X$ contained in an affine space, and a linearly reductive group $G$ acting freely on $X$, are there examples where the dimension $\dim X - \dim G$ is not realized by any of its irreducible components?
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$\begingroup$ Irreducible components of what? $\endgroup$– abxCommented Dec 9 at 10:22
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$\begingroup$ Ok, I am assuming that the quotient space is indeed a scheme. Is it too strong? $\endgroup$– User43029Commented 2 days ago
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$\begingroup$ Yes. Have a look at Mumford's Geometric Invariant Theory, chapter I. $\endgroup$– abxCommented 2 days ago
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$\begingroup$ So, I took a look at Mumford's book, but this is not quite yet what I am asking. I understand things could go wrong in the quotient and that is why he developed the theory of quotients with linearized sheaves, but what I am looking for is a concrete example where the quotient does not work, besides the classical one of C^* acting on A^n, because there, the action is not free. $\endgroup$– User43029Commented yesterday
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