Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 ?

share|improve this question

1 Answer 1

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.

share|improve this answer
    
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). –  DamienC Feb 27 '11 at 21:34

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.