Lemma 7.3.21 (0PAB) · ScalingStacks

Lemma 7.3.21. Let $\gamma_{0}$ be a minimal admissible path in $Z$ and let $z\in Z$ . We have

i([\gamma_{0}],\operatorname{id}\nolimits_{z})=\min_{\begin{subarray}{c}\gamma\text{ admiss.}\\ [\gamma]=[\gamma_{0}]\end{subarray}}i(\gamma,\operatorname{id}\nolimits_{z})=i(\gamma_{0},\operatorname{id}\nolimits_{z})=\frac{1}{2}\bigl(\sum_{c\in C(z)}(m_{c}^{+}([\gamma_{0}])+m_{c}^{-}([\gamma_{0}]))+\delta_{\gamma_{0}(0)=z}+\delta_{\gamma_{0}(1)=z}\bigr).

If $z\in Z_{o}$ , then we have

i([\gamma_{0}],\operatorname{id}\nolimits_{z})=\frac{1}{2}\bigl(\sum_{c\in C(z)^{+}}(m_{c}([\gamma_{0}])-m_{\iota(c)}([\gamma_{0}]))+\delta_{\gamma_{0}(0)=z}+\delta_{\gamma_{0}(1)=z}\bigr).

0PAC

Proof. Note that

i([\gamma_{0}],\operatorname{id}\nolimits_{z})\leq\min_{\begin{subarray}{c}\gamma\text{ admiss.}\\ [\gamma]=[\gamma_{0}]\end{subarray}}i(\gamma,\operatorname{id}\nolimits_{z})\leq i(\gamma_{0},\operatorname{id}\nolimits_{z}).

The third equality of the lemma follows from Lemma 7.1.21.

When $Z=S^{1}$ unoriented, the lemma follows from Lemma 6.2.3.

When $Z$ is a connected non-singular curve, there is an injective morphism of curves $f:Z\to S^{1}$ . We have $i([\gamma_{0}],\operatorname{id}\nolimits_{z})\geq i(f([\gamma_{0}]),\operatorname{id}\nolimits_{f(z)})=i(f(\gamma_{0}),\operatorname{id}\nolimits_{f(z)})=i(\gamma_{0},\operatorname{id}\nolimits_{z})$ , hence the first two equalities of the lemma hold for $Z$ . It follows that they hold for any non-singular curve.

Consider now a general $Z$ and let $q:\hat{Z}\to Z$ be the non-singular cover. Let $\hat{\gamma}_{0}$ be the lift of $\gamma_{0}$ to $\hat{Z}$ . We have

i(\gamma_{0},\operatorname{id}\nolimits_{z})=\sum_{\hat{z}\in f^{-1}(z)}i(\hat{\gamma}_{0},\operatorname{id}\nolimits_{\hat{z}})=\sum_{\hat{z}\in f^{-1}(z)}([\hat{\gamma}_{0}],\operatorname{id}\nolimits_{\hat{z}})\leq i([\gamma_{0}],\operatorname{id}\nolimits_{z}).

We deduce that the first two equalities of the lemma hold.

The last equality of the lemma follows from (7.3.2). ∎