Lemma 7.4.21 (0PBE) · ScalingStacks

0PBE

Lemma 7.4.21. Let $\theta$ and $\theta^{\prime}$ be two braids such that $\theta\circ\theta^{\prime}$ is a braid. We have $\deg(\theta)\cdot\deg(\theta^{\prime})\leq\deg(\theta\circ\theta^{\prime})$ .

Given $D$ a subset of $T(Z)$ containing $Z_{exc}^{+}$ and such that $D\cap\iota(D)=\emptyset$ , the following assertions are equivalent:

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$\deg(\theta)\cdot\deg(\theta^{\prime})=\deg(\theta\circ\theta^{\prime})$
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$\deg(\theta)\cdot\deg(\theta^{\prime})$ and $\deg(\theta\circ\theta^{\prime})$ have the same image in $\Gamma(Z,D)$
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$\deg(\theta)\cdot\deg(\theta^{\prime})$ and $\deg(\theta\circ\theta^{\prime})$ have the same image in $\bar{\Gamma}(Z,D)$ .

0PBF

Proof. Assume $Z=S^{1}$ unoriented. Let $M$ be a family as in §7.4.3. Assume $M$ contains $\theta_{s}(r)$ and $\theta^{\prime}_{s}(r)$ for $r\in\{0,1\}$ and all $s$ . Proposition 7.4.18 and Lemma 7.4.19 show that the inequality follows from the corresponding inequality for maps in ${\mathcal{S}}_{n}$ , which is given by Lemmas 6.2.1 and 6.2.5.

Given $Z$ a non-singular connected curve, there is an injective morphism of curves $Z\to S^{1}$ , and the lemma follows from Proposition 7.4.3 and Lemma 7.4.12. We deduce that the inequality holds for any non-singular curve $Z$ .

Consider now a general curve $Z$ and let $q:\hat{Z}\to Z$ be the non-singular cover. Since the functor $q^{\#}:\operatorname{add}\nolimits({\mathcal{P}}(Z))\to\operatorname{add}\nolimits({\mathcal{P}}(\hat{Z}))$ is compatible with degrees (Proposition 7.4.13), it follows that the inequality holds for $Z$ .

The equivalence of the three assertions follows from the fact that an element of $(\frac{1}{2}{\mathbf{Z}}_{\geq 0})^{\pi_{0}(Z)}\subset\Gamma(Z,Z_{exc}^{+})$ is zero if and only if its image in $\frac{1}{2}{\mathbf{Z}}_{\geq 0}\subset\bar{\Gamma}(Z,D)$ is zero. ∎