Enumerative geometry of elliptic curves on toric surfaces

TL;DR

This paper proves the equivalence of classical and tropical counts of elliptic curves on toric surfaces, providing explicit formulas and new proofs for curve counts in projective planes and Hirzebruch surfaces using tropical and algebraic geometry techniques.

Contribution

It establishes the equality of classical and tropical curve counts for elliptic curves on toric surfaces and derives explicit counting formulas using tropical and logarithmic Gromov-Witten theory.

Findings

Classical and tropical counts of elliptic curves are equal on toric surfaces.

A formula relating elliptic and rational curve counts on toric surfaces is derived.

A new proof of Pandharipande's elliptic curve count formula in ext{P}^2 is provided.

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 $\mathbb P^2$ 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 $\mathbb P^2$ 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 $\mathbb P^2$ with fixed $j$-invariant is obtained.