Intersection behavior of the fans L_k

Determine the intersection behavior in M^{tr}_2 of the fans associated to L_k for all k, in order to characterize how the components of the locus of curves with d-split Jacobians overlap.

Background

The locus of curves with d-split Jacobians is decomposed as the union of the subsets L_k, with k ranging over the units modulo d. Each subset is represented by a fan whose cones correspond to reduction types and whose images describe families of tropical curves in M^{tr}_2.

The paper notes that the parameterization of the split-Jacobian locus is a set description rather than a set of representatives, so different fans may encode the same or overlapping families of curves. Determining all intersections would provide a more complete description of the moduli-space structure and quantify the redundancy of the chosen parameterization.