椭圆纤维化的通用辫：从特征簇到Coxeter因子群
Universal braids for elliptic fibrations: from character varieties to Coxeter's factor groups
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中文总结 AI 辅助
该论文研究椭圆纤维化的通用可提升辫，证明S²上的椭圆纤维化无此类非平凡辫，对D²上的有限指数情形分类并关联Coxeter因子群，还将其推广至与模3整数Burau表示相关的情形。
中文摘要 AI 辅助
设π: M → B为以B = D²或B = S²为底的椭圆纤维化，其中Δ⊆B上有n个节点纤维。我们研究π的通用可提升辫：这类辫对(B,Δ)的所有坐标选择都允许存在保纤维的提升至M。当B = S²时，我们通过证明关于(S²,Δ)的SL₂特征簇的扎里斯基稠密性定理，说明不存在非平凡通用辫。当B = D²时，我们对通用辫子群在辫群Bₙ = Mod(D²,Δ)中具有有限指数的情形进行分类，并将这些例子与辫群的Coxeter因子群关联，而后者又与柏拉图立体相关。最后，我们通过考虑与任意椭圆纤维化相关的底B的标准分支覆盖族，推广有限指数情形下的结果，该推广自然将通用辫与模3约化的整数Burau表示关联起来。
英文摘要
Let $π: M \to B$ be an elliptic fibration over $B = D^2$ or $B = S^2$ with $n$ nodal fibers over $Δ\subseteq B$. We study the universal liftable braids for $π$: those braids that admit a fiber-preserving lift to $M$ for all choices of coordinates on $(B,Δ)$. When $B = S^2$, we show that nontrivial universal braids do not exist by proving a Zariski-density theorem on the $\mathrm{SL}_2$-character variety for $(S^2,Δ)$. When $B = D^2$ we classify when the subgroup of universal braids has finite index in the braid group $B_n = \mathrm{Mod}(D^2,Δ)$, and relate these examples to Coxeter's factor groups of braid groups, which in turn are related to the platonic solids. Finally, we generalize the results derived in the finite-index cases by considering a canonical family of branched covers of the base $B$ associated to any elliptic fibration. The generalization naturally connects the universal braids to the integral Burau representation reduced modulo 3.
