Abstract
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.
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Len, Y., Ranganathan, D. Enumerative geometry of elliptic curves on toric surfaces. Isr. J. Math. 226, 351–385 (2018). https://doi.org/10.1007/s11856-018-1698-9
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DOI: https://doi.org/10.1007/s11856-018-1698-9