Merge-position invariance in quadratically enriched tropical floor diagrams
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866914548317945856 |
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| author | Hedjem, Yanis |
| author_facet | Hedjem, Yanis |
| contents | Jaramillo Puentes et al. give a Grothendieck-Witt valued floor-diagram formula
for rational curves in smooth toric del Pezzo surfaces with simple and
quadratic double point conditions. We study its dependence on the choice of
merge positions, namely on which adjacent pairs of point conditions are
merged. Although independence of these choices follows abstractly from the
tropical correspondence and algebraic invariance, it is not manifest in the
floor-diagram expression.
We prove a wall-crossing factorisation for the floor formula: for any two
merge configurations, the difference is of the form $ΔN = C \prod_{j=1}
^s (\langle d_j\rangle-\langle 1\rangle)$. The coefficient $C$ admits a fixed
universal lift. Using real broccoli invariance, the possible obstruction is
reduced to a multiple of the virtual Pfister element $\langle\langle 2,d_1,
\ldots,d_s\rangle\rangle$. This gives a complete tropical proof of merge-
position invariance over every admissible field in which $2$ is a square. Over
a general admissible field, the same tropical analysis reduces the problem to
one explicit mod-$2$ congruence for the residual coefficient; this congruence
is verified by a single Laurent-series specialisation, using the tropical
correspondence of Jaramillo Puentes et al. and the algebraic invariance
theorem of Kass-Levine-Solomon-Wickelgren. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09107 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Merge-position invariance in quadratically enriched tropical floor diagrams Hedjem, Yanis Algebraic Geometry Primary 14T90, 14N10, Secondary 11E81, 14N35 Jaramillo Puentes et al. give a Grothendieck-Witt valued floor-diagram formula for rational curves in smooth toric del Pezzo surfaces with simple and quadratic double point conditions. We study its dependence on the choice of merge positions, namely on which adjacent pairs of point conditions are merged. Although independence of these choices follows abstractly from the tropical correspondence and algebraic invariance, it is not manifest in the floor-diagram expression. We prove a wall-crossing factorisation for the floor formula: for any two merge configurations, the difference is of the form $ΔN = C \prod_{j=1} ^s (\langle d_j\rangle-\langle 1\rangle)$. The coefficient $C$ admits a fixed universal lift. Using real broccoli invariance, the possible obstruction is reduced to a multiple of the virtual Pfister element $\langle\langle 2,d_1, \ldots,d_s\rangle\rangle$. This gives a complete tropical proof of merge- position invariance over every admissible field in which $2$ is a square. Over a general admissible field, the same tropical analysis reduces the problem to one explicit mod-$2$ congruence for the residual coefficient; this congruence is verified by a single Laurent-series specialisation, using the tropical correspondence of Jaramillo Puentes et al. and the algebraic invariance theorem of Kass-Levine-Solomon-Wickelgren. |
| title | Merge-position invariance in quadratically enriched tropical floor diagrams |
| topic | Algebraic Geometry Primary 14T90, 14N10, Secondary 11E81, 14N35 |
| url | https://arxiv.org/abs/2605.09107 |