2023/07/04 by Madeline Brandt, Brandt, Madeline, Melody Chan +3
Mathematics · #05E14 #14H10 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2307.01819
openalex publication_date 2023/07/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For g≥ 2 and n≥ 0, let Hg,n⊂ Mg,n denote the complex moduli stack of n-marked smooth hyperelliptic curves of genus g. A normal crossings compactification of this space is provided by the theory of pointed admissible ℤ/2ℤ-covers. We explicitly determine the resulting dual complex, and we use this to define a graph complex which computes the weight zero compactly supported cohomology of Hg, n. Using this graph complex, we give a sum-over-graphs formula for the Sn-equivariant weight zero compactly supported Euler characteristic of Hg, n. This formula allows for the computer-aided calculation, for each g≤ 7, of the generating function hg for these equivariant Euler characteristics for all n. More generally, we determine the dual complex of the boundary in any moduli space of pointed admissible G-covers of genus zero curves, when G is abelian, as a symmetric Δ-complex. We use these complexes to generalize our formula for hg to moduli spaces of n-pointed smooth abelian covers of genus zero curves.