Merge-position invariance in quadratically enriched tropical floor diagrams

TL;DR

This paper proves the invariance of a floor-diagram formula for rational curves in toric surfaces under merge-position changes, using tropical and algebraic methods, and explores implications over various fields.

Contribution

It establishes a wall-crossing factorization for the floor formula and demonstrates merge-position invariance through tropical and algebraic invariance techniques.

Findings

Proves merge-position invariance of the floor-diagram formula.

Derives a wall-crossing factorization relating different merge configurations.

Reduces the problem to a mod-2 congruence verified by Laurent-series specialization.

Abstract

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 $\Delta N = C \prod_{j=1} ^s (\langle d_j\rangle-\langle 1\rangle)$. 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.