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It is well-known, that the moduli space $\mathcal M_{1;1}$ of elliptic curves is isomorphic to an orbifold space $(S_3\times S_2) \backslash\backslash \mathcal M_{0;4}$, where the first factor of the group acts by permutation of the first three distinguished points on a rational curve, and the second one acts trivially. Let us try to exploit this idea to construct $\mathcal M_{1;2}$.

Note first, that Riemann-Roch implies that for any two (different) points $P$ and $Q$ on a genus 1 curve, there exists a degree 2 function having degree 1 poles in $P$ and $Q$. The set of critical points is the same for any such a function. It is not difficult to understand, that the set of critical values of any such a function is the same up to an automorphism of $P^1$ fixing the infinity.

Vise-versa, if one fixes four points on $\mathbb C\subset P^1$, it determines a genus one curve with two distinguished points coming as preimages of the infinity under a degree two function ramified in the chosen points.

That indicates, that there should exist an orbifold map $\mathcal M_{1;2} \to (S_4) \backslash\backslash \mathcal M_{0;5}$, that forgets the numbering of the distinguished points on a genus 1 curve. But from the other hand, any genus 1 curve with two fixed points admits an automorphism interchanging the fixed points.

So my question is the following: is the map $\mathcal M_{1;2} \to (S_4) \backslash\backslash \mathcal M_{0;5}$ an isomorphism?

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1 Answer 1

up vote 5 down vote accepted

Yes, it is.

See Leila Schneps, Special loci in moduli spaces of curves (in Galois Groups and Fundamental Groups, MSRI series 41, Cambridge University Press, 2003), pages 34-35.

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Thanks for the answer and especially for the reference. –  Max Karev Jan 16 '13 at 12:41

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