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Let's consider the moduli space $M_g$ of curves of genus $g$ over $\mathbf{C}$.

Not every curve of genus $g$ is a Galois cover (of the projective line) if $g\geq 3$.

How big is the locus of Galois covers in $M_g$? It contains the locus of cyclic covers. So there is a lower bound on the dimension ($2g-2$ if I'm not mistaken.)

Is the locus of Galois covers known to be projective or affine? (The locus of cyclic covers is known to be affine.)

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up vote 8 down vote accepted

The bigger the Galois group, the smaller the dimension. So, the biggest component is the hyperelliptic locus. There are only finitely many possibilities for the Galois group. For each fixed group, you get a quasi-projective variety, it may be affine but is definitely not projective. You can count parameters by noticing that it depends on the branch locus and that can be estimated by the Hurwitz formula.

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Is it obvious that it can never be projective? (Also, you at least need to exclude the case that the locus is a single point, like the Klein quartic in genus three.) –  Dan Petersen Feb 27 '12 at 8:15
@Dan, good point :-) It's not obvious that is not projective if positive dimensional but my feeling is that, if you can deform, you can degenerate, say, by making branch points come together. –  Felipe Voloch Feb 27 '12 at 13:37
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