Deformations of rational curves in positive characteristic
Deformations of rational curves in positive characteristic
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DOI：
10.1515/crelle-2020-0003
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发表时间：
2018-03
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影响因子：
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通讯作者：
Kazuhiro Ito;Tetsushi Ito;C. Liedtke
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文献类型：
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作者：
Kazuhiro Ito;Tetsushi Ito;C. Liedtke
摘要研究有理曲线的变形及其正特征点的奇异性。我们用它来证明，如果一个光滑的、具有正特征p的固有曲面被一组有理曲线所控制，其中一个成员具有所有的δ不变量(分别为p。雅可比数)严格小于1 2¹（p-1） {\frac{1}{2}(p-1)} (resp。p)，则曲面具有负的Kodaira维数。我们也证明了类似的，但较弱的结果适用于高维变量。此外，我们通过实例表明，我们的结果在某种意义上是最优的。在此过程中，我们用雅可比数给出了曲线在不完全域上的光滑归一化的充分判据。
Abstract 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 1 2 ( p - 1 ) {\frac{1}{2}(p-1)} (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.