Mathematics > Algebraic Geometry
arXiv:2606.21449 (math)
[Submitted on 19 Jun 2026]
Abstract:We introduce a combinatorial structure $(n,W,T)$ encoding the topological type of a curve transverse to a fixed cellular arrangement of curves on a compact real surface, in terms of intersection numbers, Dyck words and rooted trees. We apply this formalism to analyze a natural generalization of Hilbert's 16th problem to arrangements of curves. We obtain a complete classification of arrangements of three lines and a cubic, and a partial classification of arrangements of three lines and a quartic. This is achieved using Bézout-type obstructions, Viro's patchworking and translations, and by developing the Julia library NWT to handle large databases of curves.
Submission history
From: Giacomo Maletto [view email]
[v1]
Fri, 19 Jun 2026 14:08:07 UTC (7,797 KB)
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