Recall the tropical limit.
Definition 1. Let $C = \lbrace C^{(t)} \rbrace$ be a family of curves on a punctured disk given by the equation $$ f(z, w) = \sum_{(i,j) \in \Delta} a_{i,j}(t) z^i w^j, $$ where $\Delta$ is some integer polygon. From this we obtain a tropical polynomial $$ N_f(x) = \max_{(i,j) \in \Delta \cap \mathbb{Z}^2} (xi + yj + \operatorname{val}(a_{i,j})), $$ and by Legendre transform a subdivision $\Delta = \Delta_1 \cup \dotsb \cup \Delta_N$. The tropical limit of $f$ is the data of
- the tropical curve $T_f$ defined by $N_f$,
- a collection of curves $C_m \subseteq \operatorname{Tor}(\Delta_m)$ for $1 \le m \le N$ where each $C_m$ is defined by the equation $f_m(z, w) = \sum_{(i,j) \in \Delta_m \cap \mathbb{Z}^2} a_{i,j}^{(0)} z^i w^j$, where $a_{i,j}(t) = (a_{i,j}^{(0)} + O(t)) t^{\nu(i,j)}$.
Lemma 2. Let $x_1 = \operatorname{val}(p_1), \dotsc, x_\zeta = \operatorname{val}(p_\zeta)$ be $\Delta$-generic. Then the tropical limit of a curve $C \in \operatorname{Sev}_ n(\Delta)$ passing through these points $p_1, \dotsc, p_\zeta$ consists of
- a simple tropical curve $T$ with Newton polygon $\Delta$ of rank $\zeta$,
- curves $C_m \supseteq \operatorname{Tor}(\Delta_m)$ for $\Delta_m$ either a triangle or a paralleogram, satisfying the property that
- if $\Delta_m$ is a triangle then $C_m$ is a rational nodal curve crossing $\operatorname{Tor}(\partial \Delta_m)$ at exactly three points where it is unibranch,
- if $\Delta_m$ is a paralleogram then the defining equation of $C_m$ is a product of a monomial and binomials.
We now want to refine this so that we can actually recover the family from the tropical limit. To do this, we study the singularity of $C_m$ along $\operatorname{Tor}(\sigma)$ where $\sigma = \Delta_k \cap \Delta_l$.
Let us first consider the case when both $\Delta_k$ and $\Delta_l$ are triangles. Let $m$ be the lattice length of $\sigma$. By counting the intersection number, we can see that both $C_k$ and $C_l$ will be tangent to $\operatorname{Tor}(\sigma)$ (in their respective toric varieties) with multiplicity $m$, at the same point $p_\sigma \in \operatorname{Tor}(\sigma)$.
We can do a monomial coordinate transformation to make $\sigma$ be the edge connecting $(0, 0)$ and $(m, 0)$. We call the corresponding functions $f^\prime$. At this point, we let $\xi$ be the $x$-coordinate of $p_\sigma$ and then consider the function $$ f^{\prime\prime}(x-\xi, y) = f^\prime(x, y). $$ This completely destroys the monomial structure, but still has a corresponding Newton function $\nu^{\prime\prime}$ as well.
Claim 3. The function $\nu^{\prime\prime}$ is linear on the triangle $\Delta_{[\sigma]}$ with vertices $(0, -1)$ and $(0, 1)$ and $(m, 0)$. Moreover, the corresponding limit curve $C_{[\sigma]}$ of $f^{\prime\prime}$ with Newton polygon $\Delta_{[\sigma]}$ is rational.
Definition 4. We call $(\Delta_{[\sigma]}, C_{[\sigma]})$ the refinement of the tropical limit.