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Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Foliations and rational connectedness in positive characteristic

Author: Mingmin Shen
Journal: J. Algebraic Geom. 19 (2010), 531-553
Published electronically: March 8, 2010
MathSciNet review: 2629599
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Abstract | References | Additional Information

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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Additional Information

Mingmin Shen
Affiliation: Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027

Received by editor(s): May 30, 2008
Received by editor(s) in revised form: October 9, 2009
Published electronically: March 8, 2010