Enumeration of plane unicuspidal curves of any genus via tropical geometry

Abstract

We enumerate complex curves on toric surfaces of any given degree and genus, having a single cusp and nodes as their singularities, and matching appropriately many point constraints. The solution is obtained via tropical enumerative geometry. The same technique applies to enumeration of real plane cuspidal curves: We show that, for any fixed $r\ge 1$ and $d\ge 2r+3$, there exists a generic real $2r$-dimensional linear family of plane curves of degree $d$ in which the number of real $r$-cuspidal curves is asymptotically comparable with the total number of complex $r$-cuspidal curves in the family, as $d\to \infty$.

77 pages (including appendix), 9 figures; replaces preprint at arXiv:1306.3514; the main body of the paper substantially edited to match the published version

References in corpus (3)

• Refined curve counting with tropical geometry
• Patchworking singular algebraic curves, non-Archimedean amoebas and enumerative geometry
• Rational cuspidal curves on del-Pezzo surfaces

Cited by in corpus (2)

• Rational Cuspidal Curves in a moving family of ${\mathbb{P}}^2$
• On refined count of rational tropical curves