Weak approximation for rationally connected varieties over function fields

Prove that every smooth rationally connected variety over the function field of an algebraically closed curve satisfies weak approximation at all places.

Background

The paper invokes the Hassett–Tschinkel conjecture as a broad arithmetic statement about rationally connected varieties over function fields of curves. If true, it would imply weak approximation at all places for Campana sections of every Campana del Pezzo fibration with irreducible boundary, when combined with the paper's strong Campana uniruledness results and the cited weak-approximation theorem for Campana sections.

The conjecture is stronger than the specific del Pezzo-fibration results proved in the paper and remains unresolved in the stated generality.

References

The conjecture of Hassett and Tschinkel Section~1 that every smooth rationally connected variety over $K$ satisfies weak approximation at all places would imply, in combination with Theorem~\ref{thmA} and Theorem~\ref{thm:WA}, weak approximation at all places for Campana sections of every Campana del Pezzo fibration with irreducible boundary.

— Campana Rational Connectedness and Weak Approximation of Del Pezzo Orbifolds  (2609.11533 - Dandapat, 10 Sep 2026) in Remark \ref{rem:arbdegree}, Section “Weak Approximation,” subsection “Lower degree del Pezzo orbifolds”