# Moduli space of genus 1 curves with a degree n divisors

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?

• 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. Nov 30, 2020 at 8:29
• I do want them to be distinct and the divisors to be squarefree. Sorry for the confusion. Nov 30, 2020 at 15:59
• 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. Nov 30, 2020 at 16:33

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 $$p$$.

• Somewhere I saw the notation $\mathcal{M}_{1, \, [n]}$ Nov 30, 2020 at 9:39
• 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 $\Bbb{Z}/2$.
– abx
Nov 30, 2020 at 10:07
• @abx You're right, sorry! Nov 30, 2020 at 10:41
• 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...) Dec 3, 2020 at 4:56
• 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. Dec 4, 2020 at 9:32