The study of the cohomology of line bundles on (partial) flag varieties is an important problem at the intersection of algebraic geometry, commutative algebra, and representation theory. Over fields of characteristic zero, this is well-understood thanks to the Borel--Weil--Bott theorem, but in positive characteristics, it remains largely open.

In this talk, we will focus on the incidence correspondence, the partial flag variety parameterizing pairs consisting of a point in projective space and a hyperplane containing it. I will describe joint work with Claudiu Raicu, Annet Kyomuhangi, and Ethan Reed, where we derive a recursive formula for the characters of the cohomology of line bundles on the incidence correspondence in positive characteristic. In characteristic 2, we provide a closed-form formula.

I will also mention some connections with other interesting open problems and, as an application of our result, present a test for the Weak Lefschetz Property for monomial complete intersections in characteristic 2.