Genus one correspondence between tropical and algebraic curves

Alessio Cela; Sae Koyama

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arXiv · 2607.06426

Genus one correspondence between tropical and algebraic curves

Alessio Cela·Sae Koyama

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Abstract

We show that the genuinely enumerative count of algebraic elliptic curves in any toric variety agrees with the count of the corresponding well-spaced tropical curves, weighted by explicit combinatorial multiplicities. This provides a complete genus-$1$ generalization of the celebrated Nishinou--Siebert correspondence theorem in genus $0$. The proof is algebro-geometric and relies on logarithmic deformation theory together with an explicit enumeration of logarithmic maps with fixed tropicalization.

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Alessio Cela, Sae Koyama. 2026-07-07. Genus one correspondence between tropical and algebraic curves. https://arxiv.org/abs/2607.06426

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