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 \(g=1\).  Indeed, he shows that for \(n\geq p\geq 11, \mathscr{M}_{1, n, \mathbb{F}_p}\) is not uniruled... (more below the fold)