Deformations of rational curves in positive characteristic

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 is dominated by a family of rational curves such that one member has all δ-invariants (resp. Jacobian numbers) strictly les...

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Published inJournal für die reine und angewandte Mathematik Vol. 2020; no. 769; pp. 55 - 86
Main Authors Ito, Kazuhiro, Ito, Tetsushi, Liedtke, Christian
Format Journal Article
LanguageEnglish
Published De Gruyter 01.12.2020
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Summary: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 is dominated by a family of rational curves such that one member has all δ-invariants (resp. Jacobian numbers) strictly less than (resp. ), 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.
ISSN:0075-4102
1435-5345
DOI:10.1515/crelle-2020-0003