ScalingStacks

Remark 3.10. A problem when we work with the coordinates $z^{m_{1}},\ldots z^{m_{n-l}},z^{\mathfrak{m}_{j}}$ is the inequality constraint to keep $a_{m_{0}^{\prime}}e^{s\lambda(m_{0}^{\prime})}z^{m_{0}^{\prime}}$ and $a_{m_{1}^{\prime}}e^{s\lambda(m_{1}^{\prime})}z^{m_{1}^{\prime}}$ as the two dominant monomials. This means such a holomorphic chart is not quite as simple as the product of $D(1)^{\ell}$ with a long annulus in $(\mathbb{C}^{*})^{n-l}$. In practice we will cover this region by lots of simpler charts which we call the charts of boundary type. Let $P$ be any point in this region, such that $\max_{m}{|a_{m}e^{s\lambda(m)}z^{m}|}$ is large but still comparable to 1 (to guarantee the chart overlaps nontrivially with some toric type chart). The associated chart uses the same coordinates $z^{m_{1}},\ldots z^{m_{n-l}},z^{\mathfrak{m}_{j}}$ as above, but describes only a small region:

$$U_{P}=\{|z^{\mathfrak{m}_{j}}|\lesssim|z^{\mathfrak{m}_{j}}(P)|,\forall j,\quad|z^{m_{i}}-z^{m_{i}}(P)|<c|z^{m_{i}}(P)|,\forall i\},$$

where $0<c\ll 1$ is a fixed dimensional constant. These charts have an interpretation in terms of the strata $\mathcal{A}_{\lambda,\sigma}^{\infty}$ (cf. Lemma 3.5): the point $P$ corresponds roughly to a point $P^{\prime}$ on the face $F_{\sigma}^{\vee}\subset\Delta_{\lambda}^{\vee}$, and allowing $|z^{\mathfrak{m}_{j}}|$ to decrease to zero corresponds to taking the Minkowski sum with the outward normal cone $NC_{\Delta}(\sigma)$, so the tropical analogue of our small chart is $\{P^{\prime}\}+NC_{\Delta}(\Sigma)$.