亏格为\(g\)的紧致黎曼曲面上双股全辫群的有限商群与一般型复曲面：循环、二面体和超特殊商群
Finite quotients of full surface braid groups and complex surfaces of general type: cyclic, dihedral, and extra-special quotients
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中文总结 AI 辅助
研究亏格为\(g\)的紧致黎曼曲面上双股全辫群的有限循环、二面体和超特殊商群数量，在商映射特定条件下计算。并将代数结果用于构造一般型光滑曲面，得到两个三维族的极小曲面，其有相同双正则不变量和贝蒂数，但第一同调群挠部分不同。
中文摘要 AI 辅助
设\(\mathsf{B}_2(\Sigma_g)\)为亏格为\(g\)的紧致黎曼曲面上双股全辫群。在商映射\(\varphi\)不通过\(\pi_1(\operatorname{Sym^2}\Sigma_g)\)分解的假设下，我们计算了有限循环、二面体和超特殊商群\(\varphi \colon \mathsf{B}_2(\Sigma_g) \to G\)的数量。然后将代数结果应用于几何问题，即构造一般型光滑曲面作为\(\operatorname{Sym^2}(\Sigma_g)\)在对角线上分支的伽罗瓦覆盖。特别地，我们构造了两个三维族的一般型极小曲面，其\(p_g = 7\)，\(q = 4\)且\(K^2 = 32\)，不同族的成员具有相同的双正则不变量和相同的贝蒂数，但第一同调群的挠部分不同。
英文摘要
Let $\mathsf{B}_2(Σ_g)$ be the full braid group on two strings on a compact Riemann surface of genus $g$. We compute the number of finite cyclic, dihedral and extra-special quotients $φ\colon \mathsf{B}_2(Σ_g) \to G$, under the assumption that the quotient map $φ$ does not factor through $π_1(\operatorname{Sym^2}Σ_g)$. We then apply our algebraic results to the geometric problem of constructing smooth surfaces of general type as Galois covers of $\operatorname{Sym^2}(Σ_g)$ branched on the diagonal. In particular, we construct two $3$-dimensional families of minimal surfaces of general type with $p_g=7$, $q=4$ and $K^2=32$ such that members of different families have the same biregular invariants and the same Betti numbers, but different torsion part for the first homology group.
