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.)



How big is the locus of Galois covers in the moduli space of curves
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.


