ScalingStacks

Remark 8.1.9. Note that if $\alpha_{s^{\prime\prime}}\circ\zeta^{-1}$ is smooth, then the support of $\zeta$ is contained in $Z(\xi)$.

Given $\alpha$ non-zero, if $\zeta$ is positive, then both $\alpha_{s^{\prime\prime}}\circ\zeta^{-1}$ and $\zeta$ are oriented, since $\alpha_{s^{\prime\prime}}$ is oriented.