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Abstract

We prove that in a family of projective threefolds defined over an algebraically closed field, the locus of rational fibers is a countable union of closed subsets of the locus of separably rationally connected fibers. When the ground field has characteristic zero, this implies that the locus of rational fibers in a smooth family of projective threefolds is the union of at most countably many closed subfamilies.

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Acknowledgments

T. de Fernex would like to express his gratitude to Paolo Francia who first got him interested in Question 1.1; we dedicate this paper to his memory. We would like to thank Emanuele Macrì, Massimiliano Mella, and Alessandro Verra for valuable comments, and Claire Voisin for explaining to us the argument of the proof of Proposition 2.3 given below. We thank János Kollár from bringing the paper [15] to our attention, and the referees for useful comments.

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Correspondence to Tommaso de Fernex.

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T. de Fernex is partially supported by NSF CAREER Grant DMS-0847059.

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de Fernex, T., Fusi, D. Rationality in families of threefolds. Rend. Circ. Mat. Palermo 62, 127–135 (2013). https://doi.org/10.1007/s12215-013-0110-1

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  • DOI: https://doi.org/10.1007/s12215-013-0110-1

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