ScalingStacks

Proposition 3.21 . [00L4]

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Proposition 3.21.

Let n∈ℕn\in\mathbb{N} be an integer such that L⊗nL^{\otimes n} is globally generated. Let {sn,j}j∈{0,…,dn}\{s_{n,j}\}_{j\in\{0,\dots,d_{n}\}} be a basis of Vn​(L)V_{n}(L). Let ∥⋅∥n\lVert\mathord{\cdot}\rVert_{n} be a ultrametric norm on Vn​(L)V_{n}(L) with respect to which this basis is orthogonal. Let (x,e1∨​(x))∈T​o​t​(L∨)(x,e_{1}^{\vee}(x))\in Tot(L^{\vee}) and z∈(Spec⁡V∙​(L))anz\in(\spec V_{{\scriptscriptstyle\bullet}}(L))^{\mathrm{an}} be it image under p​(𝟎)anp(\boldsymbol{0})^{\mathrm{an}}, then (x,e1∨​(x))∈𝔻¯∨​(L,1n​FS​(∥⋅∥n))(x,e_{1}^{\vee}(x))\in\overline{\mathbb{D}}^{\vee}(L,\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n})) (resp.𝔻∨​(L,1n​FS​(∥⋅∥n))\mathbb{D}^{\vee}(L,\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n}))) if and only if

∀j∈{0,…,dn},|sn,j​(z)|≤∥sn,j∥n​(resp.<∥sn,j∥n).\forall j\in\{0,\dots,d_{n}\},\ \lvert s_{n,j}(z)\rvert\leq\lVert s_{n,j}\rVert_{n}\ (\text{resp.}<\lVert s_{n,j}\rVert_{n}).

In particular, the image of 𝔻∨​(L,1n​FS​(∥⋅∥n))\mathbb{D}^{\vee}(L,\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n})) under p​(𝟎)anp(\boldsymbol{0})^{\mathrm{an}} is an open subset in (Spec⁡V∙​(L))an(\spec V_{{\scriptscriptstyle\bullet}}(L))^{\mathrm{an}}.

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