ScalingStacks

Notation. (Face type regions on $X_{s}$) Consider a slightly shrinked subset of the interior of a given top dimensional face of $\partial\Delta_{\lambda}^{\vee}$. This can be regarded as a subset of $U_{w,\delta}^{\infty}\cap\partial\Delta_{\lambda}^{\vee}$, where we regard $x^{m_{i}}=s^{-1}\log|z^{m_{i}}|$ as local affine coordinates. Let $U_{w}^{s,face}\subset U_{w}^{s}$ be the subset where the $x^{m_{i}}$ coordinates correspond to points in this shrinked face. The tropical analogue of $U_{w}^{s,face}\subset U_{w}^{s}$ is the shrinked face.