The two-singularity case of the volume formula

Extend the stable-graph formula for completed Masur–Veech volumes to odd strata of quadratic differentials with exactly two singularities, including the degenerations formed by contractions of non-bridge static edges and the resulting additional terms.

Background

The main volume formula is proved under the technical assumption that the odd stratum has at least three singularities. Under this assumption, the relevant degenerations correspond to contractions of separating loops, which yield the abelian-stratum contributions described in the formula.

With exactly two singularities, additional degenerations can arise from contracting edges between distinct vertices when the associated static edges are not bridges. The authors exclude this case because these degenerations make the formulas more complicated and explicitly defer its treatment.