Every curve is a Hurwitz space in infinitely many ways - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T05:22:20Z http://mathoverflow.net/feeds/question/90912 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90912/every-curve-is-a-hurwitz-space-in-infinitely-many-ways Every curve is a Hurwitz space in infinitely many ways Harized 2012-03-11T16:26:06Z 2012-03-11T16:26:06Z <p>Diaz, Donagi and Harbater proved that every curve over $\overline{\mathbf{Q}}$ is a Hurwitz space.</p> <p>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 <code>$\mathbf{P}^1-\{0,1,\infty,\lambda\}$</code>, where $\lambda$ is allowed to vary.</p> <p>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.</p> <p>Here comes my question:</p> <p>Does there exist a <strong>natural</strong> subset $S$ of the set of Hurwitz spaces which still surjects onto the set of curves over $\overline{\mathbf{Q}}$, but with finite fibres?</p> <p>Of course, the answer is yes if we leave out <strong>natural</strong>.</p> <p>By <strong>natural</strong> I mean, for example, demanding that our Hurwitz spaces fulfill certain properties concerning the ramification type.</p>