We address the question concerning the birational geometry of the strata of holomorphic and quadratic differentials. We show strata of holomorphic and quadratic differentials to be uniruled in small genus by constructing rational curves via pencils on K3 and del Pezzo surfaces respectively. Restricting to genus 3 ≤ g ≤ 6, we construct projective bundles over a rational varieties that dominate the holomorphic strata with length at most g − 1, hence showing in addition that these strata are unirational.
We prove that the moduli space of polarized K3 surfaces of genus 11 with n marked points is unirational when n⩽6 and uniruled when n⩽7. As a consequence we settle a long standing but not proved assertion about the unirationality of M11,n for n⩽6. We also prove that the moduli space of polarized K3 surfaces of genus 11 with n⩾9 marked points has non‐negative Kodaira dimension.
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.
We show $\overline{\mathcal{M}}_{10, 10}$ and $\overline{\mathcal{F}}_{11,9}$ have Kodaira dimension zero. Our method relies on the construction of a number of curves via nodal Lefschetz pencils on blown-up $K3$ surfaces. The construction further yields that any effective divisor in $\overline{\mathcal{M}}_{g}$ with slope $<6+(12-\delta )/(g+1)$ must contain the locus of curves that are the normalization of a $\delta $-nodal curve lying on a $K3$ surface of genus $g+\delta $.
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