Uniruledness of some moduli spaces of stable pointed curves

TL;DR

This paper establishes the uniruledness of certain moduli spaces of stable pointed curves and hyperelliptic loci for specific genera and point counts, using linear systems on algebraic surfaces.

Contribution

It proves uniruledness for specific moduli spaces and hyperelliptic loci, and shows limitations on linear systems on surfaces containing general curves of certain genera.

Findings

Uniruledness of ar{M}_{g,n} for specific g and n

Uniruledness of hyperelliptic locus H_{g,n} for g  and n g+4

Restrictions on linear systems on surfaces containing general curves of genus g  12-15

Abstract

We prove uniruledness of some moduli spaces $\bar{M}_{g,n}$ of stable curves of genus $g$ with $n$ marked points using linear systems on nonsingular projective surfaces containing the general curve of genus $g$. Precisely we show that $\bar{M}_{g,n}$ is uniruled for $g=12$ and $n \leq 5$, $g=13$ and $n \leq 3$, $g=15$ and $n \leq 2$. We then prove that the pointed hyperelliptic locus $H_{g,n}$ is uniruled for $g \geq 2$ and $n \leq 4g+4$. In the last part we show that a nonsingular complete intersection surface does not carry a linear system containing the general curve of genus $g \geq 16$ and if it carries a linear system containing the general curve of genus $12 \leq g \leq 15$ then it is canonical.