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I would like to know if the moduli space $\mathcal M_{1,n}$ of genus $1$ curves with $n$ marked points can be realized as a Hurwitz space ?

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This might depend on what you count as a "Hurwitz space." To me, any cover of M_{g,n} parametrizing covers branched at the marked points is a Hurwitz space; so I would say, in a tautological tone of voice, that M_{1,n} is a Hurwitz space parametrizing degree-1 covers of elliptic curves branched at the n marked points!

But maybe you really want M_{1,n} to be a moduli space of branched covers of P^1? This seems plausible. It might be a pain to do in practice. I suppose I would try to set it up as a moduli space of covers Y -> P^1 which factor as Y -> E -> P^1, and where Y -> E is branched at the n marked points. But then you'll have to worry about collisions between marked points and Weierstrass points... it sounds like a pain.

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  • $\begingroup$ Thanks for your answer. Your guess (that I really want $M_{1,n}$ to be a moduli space of branched covers of P^1) is right. I was actually looking at arxiv.org/pdf/0802.0388 (most specifically Sections 8 and 9). My guess would have been that in type $A_n$, the Jacobi orbit space should be a moduli space of elliptic curves with marked points... but in the end the author claims that it is a Hurwitz space (in the restricted sens). $\endgroup$ – DamienC Feb 27 '11 at 21:34

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