Even-carry polynomials and cohomology of line bundles on the incidence correspondence in positive characteristic

We consider the cohomology groups of line bundles $\mathcal{L}$ on the \emph{incidence correspondence}, that is, a general hypersurface $X \subset \mathbb{P}^{n-1} \times \mathbb{P}^{n-1}$ of degrees $(1,1)$. Whereas the characteristic $0$ situation is completely understood, the cohomology in characteristic $p$ depends in a mysterious way on the base-$p$ digits of the degrees $(d, e)$ of $\mathcal{L}$. Gao and Raicu (following Linyuan Liu) prove a recursive description of the cohomology for $n = 3$, which relates to Nim polynomials when $p = 2$. In this paper, we devise a suitable generalization of Nim polynomials, which we call \emph{even-carry polynomials,} by which we can solve the recurrence of Liu--Gao--Raicu to yield an explicit formula for the cohomology for $n = 3$ and general $p$. We also make some conjectures on the general form of the cohomology for general $n$ and $p$, for which a recurrence relation was recently derived by Kyomuhangi--Marangone--Raicu--Reed.