Enumeration of Unicuspidal Curves of Any Degree and Genus on Toric Surfaces

Yaniv Ganor, Eugenii Shustin*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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 ≥ 1 and d ≥ 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 → ∞.

Original languageEnglish
Pages (from-to)16464-16523
Number of pages60
JournalInternational Mathematics Research Notices
Issue number21
StatePublished - 1 Nov 2022


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