Intersection behavior of the Lₖ-fans

Characterize the intersection behavior in M₂ᵗʳ of the fans associated with L_k for all k.

Background

For each residue class k coprime to d, the paper constructs a fan associated with L_k and maps its cones into M₂ᵗʳ. Understanding how these images intersect is necessary for determining how the various components of the d-split Jacobian locus fit together in tropical moduli space.

A complete description would also measure the redundancy of the paper’s set-theoretic parameterization of the split-Jacobian locus 𝒬, since distinct choices of k and associated reduction data may represent the same tropical curves.

References

(Can we describe)/(What is) the intersection behavior in $M{tr}_2$ of the fans associated to $L_k$ for all $k$?

— Tropical split Jacobians of curves of genus 2 II  (2502.05624 - Cobigo, 8 Feb 2025) in Section “Further Questions,” immediately after the conjecture on L₁ and L_{d−1}