Ranganathan–Wise conjecture on Gorenstein curves and smoothable differentials

Let $(C,\eta)$ be a logarithmic rubber differential with tropicalization $\bar{\lambda}$. For every level $i$, let $\lambda_i$ be the truncation of $\bar{\lambda}$, and allow a logarithmic modification $\widetilde{C}\to C$ with a natural extension $\widetilde{\eta}$ of the pullback of $\eta$.

Ranganathan–Wise conjecture. The differential $\eta$ is smoothable if and only if, for every level $i$:

1. $\lambda_i$ is a realisable tropical differential;
2. there is a reduced Gorenstein contraction

$$\sigma\colon \widetilde{C}\to\overline{C}_i$$

such that

$$\sigma^*\omega_{\overline{C}_i}=\omega_C(\lambda_i);$$

3. the differential $\widetilde{\eta}_i$ at level $i$ descends to a local generator of $\omega_{\overline{C}_i}$.

This conjecture proposes a purely algebraic characterization of smoothable multiscale differentials, replacing the global residue condition and its transcendental proof. It is stated as originally due to Ranganathan and Wise; the paper studies it in the hyperelliptic setting, where the relevant component is irreducible.