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I am sure this is well known, but I don't know what to search for:

Consider $M_{1,n}$, the moduli space of genus 1 curves with $n$ marked points. The symmetric group on $n$ letters acts on this space by acting on the marked points, and we can quotient out by this action, which corresponds to forgetting the ordering of the n points.

What is known about this space? For instance, for what $n$ (and in what characteristics) is it a rational variety? Does this space have a name? In short, how can I find literature on this?

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    $\begingroup$ Do you want the marked points to be distinct? I think the terminology "marked points" usually implies distinct points, while "degree n divisors" in the title usually would not have this restriction. $\endgroup$ Nov 30 '20 at 8:29
  • $\begingroup$ I do want them to be distinct and the divisors to be squarefree. Sorry for the confusion. $\endgroup$
    – Asvin
    Nov 30 '20 at 15:59
  • $\begingroup$ I want to say that this is related to the universal Jacobian $\mathscr{Jac}^n \to M_1$, but that should be $2$-dimensional while your space should be $(n+1)$-dimensional, so I've clearly missed something. Nevertheless, the phrase "universal Jacobian" could be useful. $\endgroup$ Nov 30 '20 at 16:33
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It doesn't have a more standard name than "$M_{1,n}/S_n$".

Belorousski's PhD thesis contains very explicit rational parametrizations of $M_{1,n}$ for $n<11$. I would expect (but haven't checked) that a modification of his constructions would work to show rationality of $M_{1,n}/S_n$, too.

The irrationality of $M_{1,11}$ can be seen from the nonvanishing Hodge number $H^0(\overline M_{1,11},\Omega^{11})$, which comes from modular forms; it is part of the two-dimensional "motive" associated to the Ramanujan cusp form $\Delta$. This motive transforms according to the sign representation of $S_n$ and in particular it dies in the cohomology of $M_{1,n}/S_n$. So I do not see an immediate cohomological obstruction for rationality. The same is true more generally that in the middle cohomology of $\overline M_{1,n}$ one finds the motive of cusp forms for $\mathrm{SL}(2,\mathbf Z)$ of weight $n+1$ and it transforms according to the sign representation. In fact according to a paper of Consani and Faber this is the only part of the rational cohomology of $\overline M_{1,n}$ transforming according to the sign representation of $S_n$.

The Kodaira dimension of $M_{1,n}$ is $0$ when $n = 11$ and $1$ for $n \geq 12$ by a paper of Bini-Fontanari. So this is at least an upper bound for the Kodaira dimension of $M_{1,n}/S_n$.

I'm not sure which (if any) parts of the above change in positive characteristic, but you could look at a paper of Will Sawin on the non-uniruledness of $\overline M_{1,n}$ in characteristic $p$.

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    $\begingroup$ Somewhere I saw the notation $\mathcal{M}_{1, \, [n]}$ $\endgroup$ Nov 30 '20 at 9:39
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    $\begingroup$ What do you mean by "the quotient of a rational variety by a reflection group is rational"? The cubic threefold is a quotient of a rational variety by $\Bbb{Z}/2$. $\endgroup$
    – abx
    Nov 30 '20 at 10:07
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    $\begingroup$ @abx You're right, sorry! $\endgroup$ Nov 30 '20 at 10:41
  • $\begingroup$ Thanks, this is very helpful. Do you think it's possible that $M_{1,n}/S_n$ is always just $\mathbb P^n$ or do you see any easy obstruction to this? (I have some computations that seem to indicate this...) $\endgroup$
    – Asvin
    Dec 3 '20 at 4:56
  • $\begingroup$ If you mean that $\overline M_{1,n}/S_n \cong \mathbb P^n$ that's certainly false (e.g. the Picard group of $\overline M_{1,2}/S_2$ has rank two). If you mean that $M_{1,n}/S_n$ is always a rational variety I highly doubt it but I don't have an argument. $\endgroup$ Dec 4 '20 at 9:32

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