Merge-position invariance in quadratically enriched tropical floor diagrams
Abstract
Jaramillo Puentes et al. define a Grothendieck--Witt-valued count by floor diagrams for rational curves in smooth toric del Pezzo surfaces with simple point conditions and quadratic double point conditions. We prove that, with the quadratic parameters kept in a fixed order, the count does not depend on which adjacent pairs form the double conditions. Given two merge configurations, we factor the difference between their floor counts as ΔN=C∏j=1s(angdj−ang1). The coefficient~C can be lifted to a free rank-three ring, independently of the field. Writing this lift as n1ang1+n2ang2+μh, invariance of the real broccoli count yields n2=−n1. The twin-tree parity condition enters through a residual calculation, which shows that n1 is even. The relation 2(ang1−ang2)=0 now makes the wall-crossing vanish. The floor count is therefore independent of merge positions over each field to which the correspondence theorem applies.
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BibTeX
@misc{hedjem2026mergeposition,
title = {{Merge-position invariance in quadratically enriched tropical floor diagrams}},
author = {Yanis Hedjem},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.09107},
eprint = {2605.09107},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.09107},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Merge-position invariance in quadratically enriched tropical floor diagrams
AU - Yanis Hedjem
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2605.09107
UR - https://arxiv.org/abs/2605.09107
AB - Jaramillo Puentes et al. define a Grothendieck--Witt-valued count by floor diagrams for rational curves in smooth toric del Pezzo surfaces with simple point conditions and quadratic double point conditions. We prove that, with the quadratic parameters kept in a fixed order, the count does not depend on which adjacent pairs form the double conditions. Given two merge configurations, we factor the difference between their floor counts as ΔN=C∏_j=1^s(angd_j−ang1). The coefficient~C can be lifted to a free rank-three ring, independently of the field. Writing this lift as n_1ang1+n_2ang2+μh, invariance of the real broccoli count yields n_2=−n_1. The twin-tree parity condition enters through a residual calculation, which shows that n_1 is even. The relation 2(ang1−ang2)=0 now makes the wall-crossing vanish. The floor count is therefore independent of merge positions over each field to which the correspondence theorem applies.
ER -