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Mathematics > Algebraic Geometry

arXiv:1807.11443 (math)
[Submitted on 30 Jul 2018 (v1), last revised 27 Aug 2021 (this version, v4)]

Title:Enumeration of plane unicuspidal curves of any genus via tropical geometry

Authors:Yaniv Ganor, Eugenii Shustin
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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 $r\ge1$ and $d\ge2r+3$, 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 $d\to\infty$.
Comments: 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
Subjects: Algebraic Geometry (math.AG)
MSC classes: 14N10, 14T05
Cite as: arXiv:1807.11443 [math.AG]
  (or arXiv:1807.11443v4 [math.AG] for this version)
  https://doi.org/10.48550/arXiv.1807.11443

Submission history

From: Eugenii Shustin [view email]
[v1] Mon, 30 Jul 2018 16:58:58 UTC (48 KB)
[v2] Fri, 12 Oct 2018 10:24:32 UTC (48 KB)
[v3] Mon, 12 Aug 2019 08:05:49 UTC (50 KB)
[v4] Fri, 27 Aug 2021 18:36:47 UTC (62 KB)
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