偶进位多项式与正特征下 incidence 对应上线丛的上同调
Even-carry polynomials and cohomology of line bundles on the incidence correspondence in positive characteristic
AI总结：
本文研究正特征下 incidence 对应上线丛的上同调，提出偶进位多项式解决 Liu-Gao-Raicu 的递推关系，得到 n=3 时的显式公式，并推测一般情况下的上同调结构。
摘要：
我们考虑 incidence 对应上线丛 𝒞 的上同调群，即一般超曲面 X ⊂ 𝔽P^{n-1} × 𝔽P^{n-1} 的度数 (1,1)。在特征 0 情况下已完全理解，但特征 p 下的上同调依赖于度数 (d,e) 的 base-p 位数的神秘方式。Gao 和 Raicu（跟随 Linyuan Liu）证明了 n=3 时的递推描述，当 p=2 时与 Nim 多项式相关。本文提出偶进位多项式作为 Nim 多项式的推广，以解决 Liu-Gao-Raicu 的递推关系，得到 n=3 时的显式公式，并推测一般情况下的上同调结构，最近由 Kyomuhangi-Marangone-Raicu-Reed 推导了递推关系。
英文摘要：
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.
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