Rational curves of submaximal quantum index
Abstract
On a complex toric surface, oriented real rational curves tangent to the toric boundary at real points are distinguished by their value of quantum index. Using variants of usual Welschinger multiplicities, G. Mikhalkin defined enumerative invariants WΔ̃κ that are provided by a signed count of oriented real rational curves with given quantum index κ and fixed tangency points on the boundary of the positive quadrant (the tangency orders being encoded by Δ̃). We prove that, for most toric surfaces and tangency orders, the invariant WΔ̃κ is zero in the case of the highest nonmaximal value of κ. This is done by enumerating tropical curves with Block--Göttsche multiplicities. Additionally, we prove that, in many situations, the lower bound, given by the absolute value |WΔ̃κ| of WΔ̃κ, for the number of oriented real rational curves is not sharp close to the tropical limit in the case of the highest nonmaximal value of κ.
Related
- Transverse rational curves and the LeBrun--Salamon conjecture
- Rational Points on Drinfeld Modular Curves of Prime Level
- On Free and Nearly Free Conjecture of Rational Unibranched Projective Plane Curves
- Hilbert schemes of low-degree rational curves on a prime Fano threefold of degree $22$
- Balancedness of Normal Bundles of Rational Curves in Grassmannians
BibTeX
@misc{pasquer2026rational,
title = {{Rational curves of submaximal quantum index}},
author = {Tangi Pasquer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41031},
eprint = {2610.41031},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41031},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Rational curves of submaximal quantum index
AU - Tangi Pasquer
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.41031
UR - https://arxiv.org/abs/2610.41031
AB - On a complex toric surface, oriented real rational curves tangent to the toric boundary at real points are distinguished by their value of quantum index. Using variants of usual Welschinger multiplicities, G. Mikhalkin defined enumerative invariants W_Δ̃^κ that are provided by a signed count of oriented real rational curves with given quantum index κ and fixed tangency points on the boundary of the positive quadrant (the tangency orders being encoded by Δ̃). We prove that, for most toric surfaces and tangency orders, the invariant W_Δ̃^κ is zero in the case of the highest nonmaximal value of κ. This is done by enumerating tropical curves with Block--Göttsche multiplicities. Additionally, we prove that, in many situations, the lower bound, given by the absolute value |W_Δ̃^κ| of W_Δ̃^κ, for the number of oriented real rational curves is not sharp close to the tropical limit in the case of the highest nonmaximal value of κ.
ER -