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.
The Kodaira dimension of the moduli space of Prym varieties