2026/07/27 by Abolfazl Soltanpour
#math.AG #math.AT #math.KT
We develop a combinatorial theory for finite arrangements of connected semialgebraic curves with ordinary multiple intersections, governed by a local node contribution ψ that determines global geometric and topological properties. We prove exact region-count formulas, characterize maximal arrangements, and extend the deletion--restriction recurrence to general curve arrangements. In the algebraic setting, we prove that the absence of triple points (kx=3) is a sufficient condition for the OS-type algebra to factor through cohomology; the converse, however, fails already for line arrangements, where the classical Orlik--Solomon relations ensure factorization even in the presence of triple points. For line arrangements we compute the discrepancy between the simplified OS-model and H2, showing it is governed by nodes with kx≥4 and equals ∑kx≥4\binomkx-12. The node contribution appears in the mixed Hodge structure via the Euler characteristic and, for line arrangements, equals dim Gr4W H2. The defect complex for concurrent lines reveals that exactness obstructions require curve-wise incidence data. Finally, we introduce binomial node invariants \Ψk\, prove Ψ2 is the universal linearly locally additive invariant, and show ψ=Ψ1-Ψ0.