Rationality in families of threefolds

TL;DR

This paper investigates the distribution of rational fibers within families of projective threefolds, showing that the rational fibers form a countable union of closed subsets, especially in characteristic zero.

Contribution

It establishes a new structural understanding of rational fibers in families of threefolds, linking rationality to separable rational connectedness and providing countability results.

Findings

Rational fibers form a countable union of closed subsets.

In characteristic zero, the rational fibers are contained in at most countably many closed subfamilies.

The results connect rationality with separable rational connectedness in threefold families.

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.