Higher-cohomology generating functions for very general surfaces of bidegree (2,e≥3) in P^1×P^2

Establish that for a very general hypersurface X ⊂ P^1 × P^2 of bidegree (2, e) with e ≥ 3, the same rational function HS(X, t_2, t_1) that generates h^0 also generates the higher cohomology series via expansions at different points, namely CS^1(X, O_X) = HS(X; t_2|_{∞}, t_1|_{0}) + HS(X; t_2|_{0}, t_1|_{∞}) and CS^2(X, O_X) = HS(X; t_2|_{∞}, t_1|_{∞}).

Background

In this surface case, Ottem’s presentation of the Cox ring yields a closed-form rational multivariate Hilbert–Poincaré series that correctly reproduces the zeroth cohomology across the entire Picard group. The conjecture strengthens this by asserting that the same rational function, when expanded about appropriately chosen combinations of zero and infinity, reproduces the first and second cohomology series.

This would demonstrate a ‘universal’ generating function encoding all line bundle cohomology dimensions on these surfaces, extending beyond the known zeroth cohomology computations from the Cox ring.