Enumerating log rational curves on some toric varieties

2025/06/16 by Carl Lian, Lian, Carl, Naufil Sakran +1 · 2 citations
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2506.13975

openalex publication_date 2025/06/16 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

The genus 0, fixed-domain log Gromov-Witten invariants of a smooth, projective toric variety X enumerate maps from a general pointed rational curve to a smooth, projective toric variety passing through the maximal number of general points and with prescribed multiplicities along the toric boundary. We determine these invariants completely for the projective bundle X=P_P^r(O^s+O(-a)), proving a conjecture of Cela--Iribar López. A different conjecture when X is the blow-up of P^r at r points is disproven. Whereas the conjectures were predicted using tropical methods, we give direct intersection-theoretic calculations on moduli spaces of "naive log quasimaps."

Citations

• Curves on Hirzebruch Surfaces and Semistability
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Cited by

• Stability of parabolic systems of Hodge bundles over punctured \mathbb P¹
• Curves in projective space and RSK

Related

• Descendant log Gromov-Witten invariants for toric varieties and tropical curves
• Higher genus Gromov-Witten invariants from projective bundles on smooth log Calabi-Yau pairs
• Tropical refined curve counting with descendants
• Counting rational curves with multiple points and Gromov-Witten invariants of blow-ups
• Logarithmic Gromov-Witten theory and double ramification cycles