Pencils on surfaces with normal crossings and the Kodaira dimension of $\overline{\mathcal{M}}_{g,n}$

TL;DR

This paper investigates the geometry of moduli spaces of pointed curves, showing certain spaces are uniruled or have non-pseudo-effective canonical divisors, and provides bounds on their Kodaira dimensions.

Contribution

It introduces new methods for smoothing pencils of curves on surfaces with normal crossings and applies these to determine the Kodaira dimension and uniruledness of specific moduli spaces.

Findings

Certain moduli spaces are uniruled.

The canonical divisor of ar{\u2113}_{g,n} is not pseudo-effective in some cases.

Bounds for the Kodaira dimension of specific moduli spaces are established.

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 pseudo-effective 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 of $(4g+5)$-pointed hyperelliptic curves $\mathcal{H}_{g,4g+5}$ is uniruled. Together with a recent result of Schwarz, this concludes the Kodaira classification for moduli of pointed hyperelliptic curves.