The fundamental group of a log terminal $$\mathbb {T}$$ T -variety

Iteration of Cox rings of klt singularities

“…Assume that Δ 𝑠 has three nontrivial coefficients. Thus, the coefficients of Δ 𝑠 correspond to platonic triples (see, e.g., [57,61]). In this case, we have that…”

“…The elements of 𝐴 0 lie in the center of 𝜋 reg 1 (𝑋, 𝑥) (see, e.g., [20,Corollary 4.13]). By [57,Example 5.1], we know that there is an exact sequence…”

“…Then, the rest of the description follows from applying Van Kampen theorem to glue the fundamental group of 𝑋(𝜎) reg × (ℙ 1 ⧵ {𝑝 1 , … , 𝑝 𝑠 }) with those of the analytic neighborhoods of the fibers of the 𝑝 𝑖 's. The proof proceeds similarly as in [57,Theorem 3.4]. □…”

“…Assume that $$\Delta _s$$ has three nontrivial coefficients. Thus, the coefficients of $$\Delta _s$$ correspond to platonic triples (see, e.g., [57, 61]). In this case, we have that $${\rm Cl}(\mathbb {P}^1,\Delta)=\langle \frac{1}{n}p,\frac{1}{m}q,\frac{1}{s}r \mid p=q=r\rangle$$.…”

“…The elements of $$A_0$$ lie in the center of $$\pi _1^{\rm reg}(X,x)$$ (see, e.g., [20, Corollary 4.13]). By [57, Example 5.1], we know that there is an exact sequence $$$$\begin{equation*} 1\rightarrow \mathbb {Z}\rightarrow \pi _1^{\rm reg}(\mathbb {P}^1,B_{\mathbb {P}^1})\rightarrow N_0 \rightarrow 1, \end{equation*}$$$$where $$N_0$$ has order at most 60. Let $$\gamma$$ be the generator of $$\mathbb {Z}\leqslant \pi _1^{\rm reg}(\mathbb {P}^1,B_{\mathbb {P}^1})$$.…”