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.)