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The kernel of the monodromy of the universal family of degree d smooth plane curves
Journal of Topology and Analysis ( IF 0.5 ) Pub Date : 2020-01-15 , DOI: 10.1142/s1793525320500375 Reid Monroe Harris 1
Journal of Topology and Analysis ( IF 0.5 ) Pub Date : 2020-01-15 , DOI: 10.1142/s1793525320500375 Reid Monroe Harris 1
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We consider the parameter space 𝒰 d of smooth plane curves of degree d . The universal smooth plane curve of degree d is a fiber bundle ℰ d → 𝒰 d with fiber diffeomorphic to a surface Σ g . This bundle gives rise to a monodromy homomorphism ρ d : π 1 ( 𝒰 d ) → Mod ( Σ g ) , where Mod ( Σ g ) : = π 0 ( Diff + ( Σ g ) ) is the mapping class group of Σ g . The main result of this paper is that the kernel of ρ 4 : π 1 ( 𝒰 4 ) → Mod ( Σ 3 ) is isomorphic to F ∞ × ℤ / 3 ℤ , where F ∞ is a free group of countably infinite rank. In the process of proving this theorem, we show that the complement Teich ( Σ g ) ∖ ℋ g of the hyperelliptic locus ℋ g in Teichmüller space Teich ( Σ g ) has the homotopy type of an infinite wedge of spheres. As a corollary, we obtain that the moduli space of plane quartic curves is aspherical. The proofs use results from the Weil–Petersson geometry of Teichmüller space together with results from algebraic geometry.
中文翻译:
d 次平滑平面曲线全族单调的核
我们考虑参数空间𝒰 d 度数的平滑平面曲线d . 度数的通用平滑平面曲线d 是一个纤维束ℰ d → 𝒰 d 具有与表面微分同胚的纤维Σ G . 这个丛产生单调同态ρ d : π 1 ( 𝒰 d ) → 模组 ( Σ G ) , 在哪里模组 ( Σ G ) : = π 0 ( 差异 + ( Σ G ) ) 是的映射类组Σ G . 本文的主要结果是内核ρ 4 : π 1 ( 𝒰 4 ) → 模组 ( Σ 3 ) 同构于F ∞ × ℤ / 3 ℤ , 在哪里F ∞ 是一个可数无限秩的自由群。在证明这个定理的过程中,我们证明了补泰克 ( Σ G ) ∖ ℋ G 超椭圆轨迹的ℋ G 在泰克米勒空间泰克 ( Σ G ) 具有无限楔形球体的同伦类型。作为推论,我们得到平面四次曲线的模空间是非球面的。证明使用了 Teichmüller 空间的 Weil-Petersson 几何的结果以及代数几何的结果。
更新日期:2020-01-15
中文翻译:
d 次平滑平面曲线全族单调的核
我们考虑参数空间