Unirationality of moduli spaces of marked elliptic curves

Question

Let us consider the moduli space of genus one curves with an effective divisor of degree d; this space is birational to the quotient of the usual moduli space of genus one curves with $d$ marked points $M_{1,d}$ by the symmetric group $S_d$ action which permutes the marked points.

I'd like to figure out whether $M_{1,d}/S_d$ is unirational for large $d$, and I am pretty sure that this must be known.

By googling the keywords I found out that $M_{1,d}$ is unirational for $d \le 10$, so that the quotient $M_{1,d}/S_d$ is unirational is well, and $M_{1,d}$ not unirational for $d \ge 11$.

My motivation to ask this is the following: in small degree, genus one curves with an effective degree $d$ divisor are obtained as complete intersections in the projective space $P^d$. For instance genus one curves with a degree $3$ divisor are parametrized by a choice of a cubic and a line in $P^2$, genus one curves with a degree $4$ divisor are intersections of two quadrics in $P^3$, and in degree $5$ one takes plane sections of the Grassmannian $Gr(2,5) \subset P^9$. This goes on for a little while (actually I am not sure for how long, perhaps for $d \le 10$), and shows that in small degree the moduli space $M_{1,d}/S_d$ is unirational. Non-unirationality in large degree will morally mean that there is no universal geometric construction of genus one curves with a degree $d$ divisor.

Answer 1

Every degree $d$ divisor is a section of a line bundle of degree $d$, which we can write as $\mathcal O(d P)$ for one of $d^2$ possible base points $P$.

So the moduli space of genus $1$ curves with a degree $d$ divisor can be covered by the moduli space of genus one curves with a marked point $P$ and a nonzero section, up to scaling, of $\mathcal O(dP)$, which is covered by a $\mathbb P^{d-1}$ bundle on $\mathbb P^1$ obtained by taking some universal family of elliptic curves (well-defined up to quadratic twist) and looking at the sections of the $d$th power of the standard line bundle.