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arXiv · 0906.0818

Foliations and Rational Connectedness in Positive Characteristic

Mingmin Shen
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Abstract

In this paper, the technique of foliations in characteristic $p$ is used to investigate the difference between rational connectedness and separable rational connectedness in positive characteristic. The notion of being freely rationally connected is defined; a variety is freely rationally connected if a general pair of points can be connected by a free rational curve. It is proved that a freely rationally connected variety admits a finite purely inseparable morphism to a separably rationally connected variety. As an application, a generalized Graber-Harris-Starr type theorem in positive characteristic is proved; namely, if a family of varieties over a smooth curve has the property that its geometric generic fiber is normal and freely rationally connected, then it has a rational section after some Frobenius twisting. We also show that a freely rationally connected variety is simply connected.

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Mingmin Shen. 2009-10-17. Foliations and Rational Connectedness in Positive Characteristic. https://arxiv.org/abs/0906.0818

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