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.

中文翻译：

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具有法线交叉的表面上的铅笔和小平尺寸

我们研究了法线交叉曲面上曲线铅笔的平滑。因此，我们证明了$\overline {\mathcal {M}}_{g,n}$在某些范围内不是伪有效的，这意味着$\overline {\mathcal {M}}_{12,6}$,$\overline {\mathcal {M}}_{12,7}$,$\overline {\mathcal {M}}_{13,4}$和$\overline {\mathcal {M}}_{14,3}$是不规则的。我们提供了 Kodaira 维度的上限$\overline {\mathcal {M}}_{12,8}$和$\overline {\mathcal {M}}_{16}$. 我们还证明了模空间$(4g+5)$尖的超椭圆曲线$\overline {\mathcal {H}}_{g,4g+5}$是不规则的。连同 Schwarz 最近的一个结果，这总结了具有负 Kodaira 维数的尖形超椭圆曲线的模量分类。