Uniruledness of some moduli spaces of stable pointed curves

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 $n\le 5$, $g=13$ and $n\le 3$, $g=15$ and $n\le 2$. We then prove that the pointed hyperelliptic locus $H_{g,n}$ is uniruled for $g\ge 2$ and $n\le 4g+4$. In the last part we show that a nonsingular complete intersection surface does not carry a linear system containing the general curve of genus $g\ge 16$ and if it carries a linear system containing the general curve of genus $12\le g\le 15$ then it is canonical.

final version, minor improvements in the exposition

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• The pseudo-effective cone of a compact Kähler manifold and varieties of negative Kodaira dimension
• The geometry of the moduli space of odd spin curves
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• Rational parametrizations of moduli spaces of curves
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Cited by in corpus (3)

• On the Kodaira dimension of the moduli space of hyperelliptic curves with marked points
• Pencils on surfaces with normal crossings and the Kodaira dimension of ${\overline{\mathcal{M}}}_{g,n}$
• Birational Geometry of moduli spaces of pointed curves