ScalingStacks

Proposition 3.21. Let $n\in\mathbb{N}$ be an integer such that $L^{\otimes n}$ is globally generated. Let $\{s_{n,j}\}_{j\in\{0,\dots,d_{n}\}}$ be a basis of $V_{n}(L)$. Let $\lVert\mathord{\cdot}\rVert_{n}$ be a ultrametric norm on $V_{n}(L)$ with respect to which this basis is orthogonal. Let $(x,e_{1}^{\vee}(x))\in Tot(L^{\vee})$ and $z\in(\spec V_{{\scriptscriptstyle\bullet}}(L))^{\mathrm{an}}$ be it image under $p(\boldsymbol{0})^{\mathrm{an}}$, then $(x,e_{1}^{\vee}(x))\in\overline{\mathbb{D}}^{\vee}(L,\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n}))$ (resp.$\mathbb{D}^{\vee}(L,\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n}))$) if and only if

In particular, the image of $\mathbb{D}^{\vee}(L,\frac{1}{n}\mathrm{FS}(\lVert\mathord{\cdot}\rVert_{n}))$ under $p(\boldsymbol{0})^{\mathrm{an}}$ is an open subset in $(\spec V_{{\scriptscriptstyle\bullet}}(L))^{\mathrm{an}}$.