Diaz, Donagi and Harbater proved that every curve over $\overline{\mathbf{Q}}$ is a Hurwitz space.

A Hurwitz space is a connected component of the curve $H_n$. The curve $H_n$ is (the compactification of) the moduli space of covers of $\mathbf{P}^1$ unramified over $\mathbf{P}^1-\{0,1,\infty,\lambda\}$, where $\lambda$ is allowed to vary.

The natural map from the set of Hurwitz spaces (up to isomorphism) to the set of smooth projective connected curves over $\overline{\mathbf{Q}}$ is surjective (by Diaz-Donagi-Harbater), but with infinite fibres.

Here comes my question:

Does there exist a natural subset $S$ of the set of Hurwitz spaces which still surjects onto the set of curves over $\overline{\mathbf{Q}}$, but with finite fibres?

Of course, the answer is yes if we leave out natural.

By natural I mean, for example, demanding that our Hurwitz spaces fulfill certain properties concerning the ramification type.


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