Computing the cohomology of line bundles on the incidence correspondence and related invariants
Annet Kyomuhangi, Emanuela Marangone, Claudiu Raicu, Ethan Reed
TL;DR
The paper introduces IncidenceCorrespondenceCohomology, a Macaulay2 package that implements KMRR's formulas for the cohomology of line bundles on the incidence correspondence $X\subset \mathbf{P}\times\mathbf{P}^\vee$, including characteristic-dependent phenomena. It provides recursive and non-recursive methods to compute $h^i(D^d\mathcal{R}(e))$ and $h^i(\mathcal{O}_X(a,b))$, as well as tools for splitting types of principal parts and for exploring the graded Han–Monsky representation ring. Key contributions include the recursiveDividedCohomology and nimDividedCohomology methods, incidenceCohomology for $X$, and splittingFdr/splittingPrincipalParts for principal parts, combined with a suite of HM-based Lefschetz tests (hasWLP/hasSLP) and generators for WLP-failing monomial complete intersections. The work advances computational techniques in positive characteristic, enabling explicit character computations, equivariant splittings, and Lefschetz analysis, with demonstrations and performance comparisons illustrating speedups and practical applicability. Overall, it integrates cohomology, representation theory, and invariant-theoretic approaches to support research in algebraic geometry and commutative algebra in varied characteristics.
Abstract
We describe the package "IncidenceCorrespondenceCohomology" for the computer algebra system Macaulay2. The main feature concerns the computation of characters and dimensions for the cohomology groups of line bundles on the incidence correspondence (the partial flag variety parametrizing pairs consisting of a point in projective space and a hyperplane containing it). Additionally, the package provides tools for (1) computing the multiplication in the graded Han-Monsky representation ring, (2) determining the splitting type of vector bundles of principal parts on the projective line, and (3) testing the weak and strong Lefschetz properties for Artinian monomial complete intersections.