A variety satisfies the local-to-global principle for rational points if the existence of local points at all places of Q implies the existence of (global) rational points over Q. In this talk, we consider a local-to-global principle for rationality (i.e. the property of being birational to projective space). We construct an example of a smooth projective threefold that violates this local-to-global principle: it is irrational over Q, but it is rational at all places. Furthermore, its reduction mod p is rational for all p ≠ 2. Our example is a complete intersection of two quadrics in P^5, and we show it has the desired rationality behavior by constructing an explicit element of order 4 in the Tate–Shafarevich group of the Jacobian of an associated genus 2 curve. This work is joint with Sarah Frei.
A threefold violating a local-to-global principle for rationality
Lena Ji, University of MichiganAuthors: Sarah Frei, Lena Ji
2023 AWM Research Symposium
Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]