Elliptic curve counting in toric threefolds: virtual, enumerative, and tropical

2026/08/06 by Sae Koyama
Mathematics · #math.AG

paper · pdf

36 pages, 19 figures. Comments welcome

arxiv created 2026/08/06 · arxiv updated 2026/08/07

Abstract

We study the enumerative geometry of elliptic curves in toric threefolds. We consider enumerative integer invariants, called well-spaced counts, which can be studied using well-spaced genus-one tropical curves in ℝ³. By comparing this with the logarithmic degeneration formula, we obtain an explicit relationship between logarithmic virtual invariants and these geometric invariants. The result is a logarithmic analogue of a formula of Getzler--Pandharipande for elliptic curves in ℙ³. As an application, we show that the virtual logarithmic invariants for ℙ³ with respect to its toric boundary are strictly less than the ordinary Gromov--Witten invariants once the degree is sufficiently large. Several examples are included.

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