Year: 2026
Type: preprint
Abstract: We study higher complements of pseudohyperplane arrangements and the specialization maps between them, induced by degenerations. We develop a presheaf formalism on oriented matroids and weak maps, which model degenerations. Our main constructions are the posets of weighted and weightwise topes, which provide functorial combinatorial models for higher complements and their specialization morphisms. Their presheaf structure yields a deletion-contraction homotopy pushout, from which we derive a uniform combinatorial proof of the integral cohomology presentation for all $n \geq 2$. This presentation is natural with respect to weak maps and identifies cohomology rings with higher Orlik-Solomon algebras extending the known formulas for linear arrangements to non-realizable oriented matroids. Finally, we equip Deligne groupoids with a presheaf structure and identify them functorially with the fundamental groupoids of the complexified complements.
Source: arXiv (Cornell University)
Authors: Coron, Basile, Lyskov, Denis
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223673120