Manin's conjecture for smooth cubic surfaces

Prove Manin's conjecture for rational points on smooth cubic surfaces, including the predicted asymptotic for points of bounded height after removal of the appropriate exceptional thin set.

Background

The paper places del Pezzo orbifolds of degree at most five in the context of Manin-type conjectures for Campana points. It notes that even the corresponding conjecture for ordinary rational points on low-degree del Pezzo surfaces is unresolved in general.

Smooth cubic surfaces are identified as a prominent remaining case, and the classified del Pezzo orbifolds are proposed as non-toric test cases for logarithmic and orbifold versions of the conjecture.

References

Even for rational points, Manin's conjecture for smooth del Pezzo surfaces of degree at most five is known only in special cases (see, e.g., ) and remains open for smooth cubic surfaces.

— Classification of low degree del Pezzo orbifolds II  (2610.06357 - Dandapat, 5 Oct 2026) in Section 1, subsection “Campana points of bounded height”

The main bottleneck in obtaining the expected exponent $\gamma_{a,b}=1$ is that we do not have sharp bounds for the number of rational points of bounded height on smooth cubic surfaces.

— Towards linear growth for biprojective hypersurfaces of small dimension  (2610.13163 - Browning et al., 9 Oct 2026) in Section 1, Introduction, immediately after Theorem 1.1