Intersection behavior of all split-Jacobian fans

Characterize the intersection behavior in \(M^{tr}_2\) of the fans associated to \(L_k\) for all admissible values of \(k\).

Background

The paper uses the fans associated with the loci $L_k$ to organize the tropical moduli locus of curves with $d$-split Jacobians. Because the description of $\mathcal{Q}$ is given as a parametrized set rather than as a set of representatives, distinct $L_k$ may map to overlapping regions of $M^{tr}_2$.

Understanding these intersections would clarify the global structure of $\mathbb{T}\mathcal{L}_d$ and quantify the redundancy in the working parametrization of the split-Jacobian locus.