# ag.algebraic geometry – Is the irreducible locus of the personality selection a principal bundle in Zariski topology? Answer

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ag.algebraic geometry – Is the irreducible locus of the personality selection a principal bundle in Zariski topology?

First, let’s occupy that the genus $$g$$ of $$Sigma$$ is bigger than or equal to 2 (in any other case the irreducible locus energy breathe vacant if $$G$$ is non-abelian).

Then for many decisions of $$G$$, the respond is not any, since there are irreducible representations which have centralizers bigger than the focus of $$G$$ (these are known as “bad representations”).

To narrative for this, you need to prohibit to the so-called “good locus”, that’s, the clique of representations whose centralizers are equal to the focus of the $$G$$.

In that illustration, I consider an dispute much like Lemma 2.2, given Theorem 3, ought to display that the map $$Hom^{good}(pi_1(Sigma),G)to Hom^{good}(pi_1(Sigma),G)//G$$ is a $$PG$$-bundle within the analytic topology ($$PG=G/Z(G)$$).

I anticipate that the motion of $$PG$$ is in actual fact scheme-theoretically free on the nice locus and thus, by the same dispute as Corollary 2.2.8, the map ought to too breathe a $$PG$$-bundle within the sense of Definition 4.8 (étale domestically trifling).

I do not know off the highest of my head if there are instances the place these bundles are domestically trifling within the Zariski topology, however I extremely doubt it. Some anecdotal proof for that is the evolution of instruments to deal with fibrations on orbit-type strata of illustration/personality varieties which can be domestically trifling within the analytic topology however not the Zariski topology for computing E-polynomials (behold right here for instance).

Lastly, delight settle for my apologetic for this terse and uneven “answer”. I obtained an e mail notification about this query and thought I’d give a brisk response off the highest of my head (I’m not actually taking part in MO today). I await it helps (at the least to offer you some path). Feel free to e mail me you probably have questions (I’ll not breathe checking MO).

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