Pencils on Surfaces with Normal Crossings and the Kodaira Dimension of $\overline {\mathcal {M}}_{g,n}$

Abstract

We study smoothing of pencils of curves on surfaces with normal crossings. As a consequence we show that the canonical divisor of $\overline {\mathcal {M}}_{g,n}$ is not pseudoeffective in some range, implying that $\overline {\mathcal {M}}_{12,6}$, $\overline {\mathcal {M}}_{12,7}$, $\overline {\mathcal {M}}_{13,4}$ and $\overline {\mathcal {M}}_{14,3}$ are uniruled. We provide upper bounds for the Kodaira dimension of $\overline {\mathcal {M}}_{12,8}$ and $\overline {\mathcal {M}}_{16}$. We also show that the moduli space of $(4g+5)$-pointed hyperelliptic curves $\overline {\mathcal {H}}_{g,4g+5}$ is uniruled. Together with a recent result of Schwarz, this concludes the classification of moduli of pointed hyperelliptic curves with negative Kodaira dimension.

Keywords

• 14H10
• 14H45
• 14M20