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Rational connectivity and sections of families over curves

T. Graber·J. Harris·B. Mazur·J. Starr
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Abstract

We show that if a family of complex varieties over a base B admits a section when restricted to a very general curve in B, then the family must contain a subfamily of rationally connected varieties dominating B. As an application, we deduce the existence of one parameter families of Enriques surfaces with no section.

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T. Graber, J. Harris, B. Mazur, J. Starr. 2002-10-15. Rational connectivity and sections of families over curves. https://arxiv.org/abs/math/0210225

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