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arXiv · 2508.05555

From arcs to curves: quadratic growth of 1-systems

Tarik Aougab·Jonah Gaster
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Abstract

We show that a collection of simple closed curves pairwise intersecting at most once on an orientable surface of Euler characteristic $χ$ has at most $2016|χ|^2+338|χ|$ curves. Up to multiplicative constants, this resolves a thirty-year old problem (see Problem 2.12(b) from the K3 Problem List). Inspired by the work of Przytycki in the setting of arcs, we introduce the concepts of tulips, flowers, and stem systems in order to account for how certain polygons built from pairs of curves in the collection distribute area over the surface.

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Tarik Aougab, Jonah Gaster. 2026-09-17. From arcs to curves: quadratic growth of 1-systems. https://arxiv.org/abs/2508.05555

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