We study deformations of rational curves and their singularities in positive characteristic. We use this to prove that if a smooth and proper surface in positive characteristic p is dominated by a family of rational curves such that one member has all δ-invariants (resp. Jacobian numbers) strictly less than $\frac{1}{2}\left(p-1\right)$ (resp. p), then the surface has negative Kodaira dimension. We also prove similar, but weaker results hold for higher-dimensional varieties. Moreover, we show by example that our result is in some sense optimal. On our way, we obtain a sufficient criterion in terms of Jacobian numbers for the normalization of a curve over an imperfect field to be smooth.

中文翻译：

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正特性曲线的有理变形

我们以正特性研究有理曲线的变形及其奇异性。我们用它来证明，如果正特性p上的光滑且适当的表面被一族有理曲线所控制，则一个成员的所有δ不变量（分别为雅可比数）严格小于$\frac{\mathrm{1个}}{2}（p--\mathrm{1个}）$（分别为p），则该表面的Kodaira尺寸为负。我们也证明了类似的结果，但对于高维品种则结果较弱。此外，我们通过示例表明我们的结果在某种意义上是最优的。在我们的路上，我们获得了关于雅可比数的足够标准，可以使不完美区域上的曲线归一化以使其平滑。