Enumerative geometry of elliptic curves on toric surfaces
Date
06/2018Metadata
Show full item recordAbstract
We establish the equality of classical and tropical curve counts for elliptic curves on toric surfaces with fixed j-invariant, refining results of Mikhalkin and Nishinou--Siebert. As an application, we determine a formula for such counts on ℙ2 and all Hirzebruch surfaces. This formula relates the count of elliptic curves with the number of rational curves on the surface satisfying a small number of tangency conditions with the toric boundary. Furthermore, the combinatorial tropical multiplicities of Kerber and Markwig for counts in ℙ2 are derived and explained algebro-geometrically, using Berkovich geometry and logarithmic Gromov--Witten theory. As a consequence, a new proof of Pandharipande's formula for counts of elliptic curves in ℙ2 with fixed j-invariant is obtained.
Citation
Len , Y & Ranganathan , D 2018 , ' Enumerative geometry of elliptic curves on toric surfaces ' , Israel Journal of Mathematics , vol. 226 , pp. 351–385 . https://doi.org/10.1007/s11856-018-1698-9
Publication
Israel Journal of Mathematics
Status
Peer reviewed
ISSN
0021-2172Type
Journal article
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