Uniruledness of some moduli spaces of stable pointed curves (Q392431)

The author shows uniruledness of the moduli spaces \(\overline{\mathcal{M}}_{g,n}\) of pointed stable curves for \(g = 12\) and \(n \leq 5\), \(g = 13\) and \(n \leq 3\), as well as \(g = 15\) and \(n \leq 2\), improving on earlier known bounds in all cases. A general result is proven stating that if \(C\) is a general curve of genus \(g \geq 2\) moving on a nonsingular projective regular surface \(S\) in a non-isotrivial linear system of projective dimension \(r\) whose deformations are sufficiently general, then \(\overline{\mathcal{M}}_{g,n}\) is uniruled for \(n \leq r + \rho\), where \(\rho = \rho(g, r, d)\) is the Brill-Noether number. This theorem is then applied to the specific geometry of the cases \(g = 12, 13, 15\) by building on constructions by \textit{A. Bruno}, \textit{M. C. Chang}, \textit{Z. Ran} and \textit{A. Verra}. The surfaces \(S\) used in the proof are a quintic in \(\mathbb{P}^3\) for \(g = 13\) and a complete intersection of four quadrics in \(\mathbb{P}^6\) for \(g = 12\) and \(15\).NEWLINENEWLINEThe method can also be applied to show uniruledness of suitable subloci of \(\overline{\mathcal{M}}_{g,n}\). As an example the author shows that the loci \(\mathcal{H}_{g,n}\) of pointed hyperelliptic curves are uniruled for all \(g \geq 2\) and \(n \leq 4g + 4\). These were recently shown by \textit{G. Casnati} to be even rational when \(g \geq 3\) and \(n \leq 2g + 8\).NEWLINENEWLINEIn the final part of the paper it is shown that in applying the main theorem, the surface \(S\) can be chosen as a complete intersection only when \(g \leq 15\), thus making it harder to construct suitable surfaces for higher genera.