Zero-cycles over a function field in nets of double planes

Let d ≥ 3 and let (F₀, F₁, F₂) be a very general ordered triple of plane forms of degree 2d, with F₁ = F₂ = 0 transverse and disjoint from F₀ = 0. For the smooth double plane F over K = ℂ(t,u) defined by y² = F₀ + tF₁ + uF₂, we compute the entire rational Chow group of zero-cycles: it has dimension 4d² + 1, with degree-zero basis given by the signed pairs above the 4d² base points. The main step determines the full vertical subgroup in a smooth projective incidence model. Its anti-invariant part vanishes for very general triples, by signed parameter monodromy, proper parameter spaces for cycles, and specialization to Mumford's theorem. We also construct closed points of residue degree 2d − 1 with prescribed opposite basis classes, and identify the corresponding lowest-weight cohomology with an explicit compact correspondence return. The compact motive is an instance of the classical Cayley trick. The zero-cycle calculation concerns K, and makes no finite-dimensionality assertion for geometric generic or complex fibers. MSC 2020: Primary 14C25; Secondary 14D05, 14J28, 14C30.