Weak approximation for low-degree del Pezzo surfaces

Establish weak approximation at every place for smooth del Pezzo surfaces of degree at most two over the function field of an algebraically closed curve of characteristic zero.

Background

The paper recalls that weak approximation is known for smooth del Pezzo surfaces of degree at least four over the function field of a curve, and for cubic surfaces under additional results cited in the paper. The general low-degree cases are relevant because weak approximation of the generic fibre is used to deduce weak approximation for Campana sections of del Pezzo fibrations.

The unresolved range explicitly identified in the paper is degree at most two. The paper records several special cases where weak approximation is known, including square-free discriminant and sufficiently generic families, but not the general statement.

References

For $d \leq 2$ the corresponding statement is open in general \S~2.

— Campana Rational Connectedness and Weak Approximation of Del Pezzo Orbifolds  (2609.11533 - Dandapat, 10 Sep 2026) in Section “Weak Approximation,” immediately following Theorem 4.1 (Theorem \ref{thm:CTG})

Colliot-Thélène conjectured that all such failures are explained by the Brauer-Manin obstruction (see ).

— Classification of low degree del Pezzo orbifolds II  (2610.06357 - Dandapat, 5 Oct 2026) in Section 1, subsection “Local-global principles and special orbifolds”