One of my favorite questions is: for which \(g, n, p\) is the moduli space of \(n\)-pointed genus \(g\) curves \(\mathscr{M}_{g,n, \mathbb{F}_p}\) unirational/uniruled? Will Sawin has just posted a beautiful paper on the ArXiv answering this question in most cases, for \(g=1\). He shows that for \(n\geq p\geq 11, \mathscr{M}_{1, n, \mathbb{F}_p}\) is not uniruled.
Severi believed that \(\mathscr{M}_{g, n}\) was unirational (or possibly even rational) for all \(g, n\) (some call this "Severi's conjecture"). This belief was shown to be false in characteristic zero by Harris and Mumford, who showed that for \(g\geq 24\), \(\mathscr{M}_{g,n}\) is of general type. We now know quite a bit about the birational geometry of the moduli of curves.
But being general type does not rule out unirationality/uniruledness in positive characteristic, because it does not rule out e.g. inseparable maps from varieties with negative Kodaira dimension! So in principle, Severi's conjecture could have been true in positive characteristic. Will shows that this is not the case.
Sawin first defines a birational invariant (which is related to an older invariant defined by Ekedahl) as follows; if \(X\) is a \(d\)-dimensional variety over a finite field \(\mathbb{F}_q\), he defines \(H_{tdF}^*(X)\) to be the quotient of \(H^*_c(X_{\overline{\mathbb{F}_q}}, \mathbb{Q}_\ell)\) by the maximal Frobenius-stable subspace on which the eigenvalues of Frobenius divide \(q^{d-1}\) in the ring of algebraic integers. Here "tdF" stands for "top-dimensional Frobenius" --- this is some notion of primitive cohomology, in that it cannot come from proper subvarieties. Moreover, because of the existence of trace maps on etale cohomology, dominant rational maps induce injections on \(H^*_{tdF}\). As projective space has trivial \(H^*_{tdF}\), we see immediately that if \(H^*_{tdF}(X)\not=0\), \(X\) is not unirational. (And one may similarly rule out uniruledness by showing that \(H^d_{tdF}(X)\not=0\).)
So how does one show that \(H^*_{tdF}(\overline{\mathscr{M}_{1,11}})\) is non-zero? Essentially the idea is that cusp forms give cohomology classes on \(\mathscr{M}_{1,n}\) of level 1 and weight less than \(n+2\) give cohomology classes on the desired moduli space --- one may then compute the eigenvalues of Frobenius on these classes by Eichler-Shimura. Sawin then directly computes to show that there are eigenforms with the desired properties.
It is likely that a similar method (with substantially more involved computations) would give a proof of non-uniruledness for \(\mathscr{M}_{g,n}, n\gg p\gg 0\) for \(g\leq 3\). But (as Sawin points out), it's unlikely to generalize to \(g\gg 0\), since any class in \(H_{tdF}\) is non-tautological, and hence somewhat hard to study in cases where the cohomology of \(\mathscr{M}_g\) is not accessible by automorphic means.
Sawin also also suggests that there may be large \(n\) such that \(M_{1, n}\) is uniruled in characteristics \(2,3,5,7\). If true, this would be awesome! If not, how would one prove it?