Enumerative geometry of elliptic curves on toric surfaces

Y. Len, D. Ranganathan

Research output: Contribution to journalArticlepeer-review

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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.
Original languageEnglish
Pages (from-to)351–385
JournalIsrael Journal of Mathematics
Volume226
Early online date11 May 2018
DOIs
Publication statusPublished - Jun 2018

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