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.
Classification of smooth factorial affine surfaces of Kodaira dimension zero with trivial units
