# Rationality in families of threefolds

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Davide Fusi, University of Utah
Fine Hall 322

In a joint work with Tommaso de Fernex, 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 a countable union of closed subsets of the parameter space. General expectation suggests that the result maybe false in higher dimension.