Universal generating functions for all line bundle cohomologies on hypersurfaces in P^1×P^n

Establish that for a general hypersurface X of bidegree (d,e) in P^1 × P^{n} with either d ≤ n (and e arbitrary) or e = 1 (and d arbitrary), the single rational function derived from the Cox ring encodes all nonzero line bundle cohomology series by expansion at specified points—yielding CS^0(X, O_X), CS^1(X, O_X), CS^{n−1}(X, O_X), and CS^{n}(X, O_X)—and that all intermediate cohomology groups vanish.

Background

For hypersurfaces in P^1×P^n within the Mori dream space regime (d ≤ n or e = 1), the Cox ring has an explicit presentation, allowing the authors to write a rational Hilbert–Poincaré series that generates h^0. The conjecture posits that this same rational function, when expanded around different combinations of zero and infinity in the counting variables, yields all higher nonvanishing cohomology series, with all intermediate cohomologies identically zero.

This would generalize the ‘universal’ generating-function phenomenon observed in lower-dimensional cases, providing a comprehensive and exact encoding of line bundle cohomology across Fano, Calabi–Yau, and general-type examples in this class.