ScalingStacks

Lemma 2.13 . [059R]

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Lemma 2.13.

In the situation of Proposition 2.11 let τ\tau be an open face of the skeleton Δ\Delta of dimension equal to the dimension of 𝔛′a​n\mathfrak{X}^{\prime an} and assume that hh is affine linear on τ¯\bar{\tau}. Let DD be the induced Cartier divisor on 𝔛′′\mathfrak{X}^{\prime\prime} and YY a proper curve in 𝔛~′′\tilde{\mathfrak{X}}^{\prime\prime} with Y⊆red𝔛′′⁡(p𝔛′′−1​(τ))Y\subseteq\red_{\mathfrak{X}^{\prime\prime}}(p_{\mathfrak{X}^{\prime\prime}}^{-1}(\tau)) e.g. if YY lies inside an irreducible component of 𝔛~′′\tilde{\mathfrak{X}}^{\prime\prime} corresponding to a vertex u∈τu\in\tau of 𝔇\mathfrak{D}. Then deg(D.Y)=0\Deg(D.Y)=0.

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