Enumerative geometry of elliptic curves on toric surfaces

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.

v3: 23 pages, 11 TikZ figures. Several details added to clarify Section 3, other minor changes. To appear in the Israel Journal of Mathematics

References in corpus (6)

• Enumerative tropical algebraic geometry in R2
• Logarithmic Geometry and Moduli
• Skeletons of stable maps I: Rational curves in toric varieties
• Faithful realizability of tropical curves
• Genus one enumerative invariants in del-Pezzo surfaces with a fixed complex structure
• Stable logarithmic maps to Deligne-Faltings pairs I

Cited by in corpus (5)

• Moduli of stable maps in genus one and logarithmic geometry II
• Curve counting in genus one: elliptic singularities & relative geometry
• Kirchhoff's theorem for Prym varieties
• Genus one enumerative invariants in del-Pezzo surfaces with a fixed complex structure
• Genus two enumerative invariants in del-Pezzo surfaces with a fixed complex structure