Pencils on Surfaces with Normal Crossings and the Kodaira Dimension of

Abstract

AbstractWe 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.