  For a finite dimensional vector space V of dimension n, we consider the
incidence correspondence (or partial flag variety) X in P(V) x P(V*),
parametrizing pairs consisting of a point and a hyperplane containing it. We
completely characterize the vanishing and non-vanishing behavior of the
cohomology groups of line bundles on X in characteristic p>0. If n=3 then X is
the full flag variety of V, and the characterization is contained in the thesis
of Griffith from the 70s. In characteristic 0, the cohomology groups are
described for all V by the Borel-Weil-Bott theorem. Our strategy is to recast
the problem in terms of computing cohomology of (twists of) divided powers of
the cotangent sheaf on projective space, which we then study using natural
truncations induced by Frobenius, along with careful estimates of
Castelnuovo-Mumford regularity. When n=3, we recover the recursive description
of characters from recent work of Linyuan Liu, while for general n we give
character formulas for the cohomology of a restricted collection of line
bundles. Our results suggest truncated Schur functions as the natural building
blocks for the cohomology characters.
