The rational Chow rings of moduli spaces of hyperelliptic curves with marked points

Samir, Canning,; Larson, Hannah

doi:10.48550/arxiv.2207.10873

Cited by 1 publication

(1 citation statement)

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“…Recently, there have been a number of significant advances on the geometry of moduli spaces of pointed hyperelliptic curves. Canning-Larson study the rational Chow ring of H 𝑔,𝑛 -in particular, determining it completely for 𝑛 ≤ 2𝑔 + 6 [16]. Their results also have implications for rationality of H 𝑔,𝑛 .…”

Section: Related Work On the Cohomology Of H 𝑔𝑛mentioning

confidence: 97%

On the weight zero compactly supported cohomology of

Brandt,

Chan,

Kannan

2024

Forum of Mathematics, Sigma

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For $g\ge 2$ and $n\ge 0$ , let $\mathcal {H}_{g,n}\subset \mathcal {M}_{g,n}$ 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 $\mathbb {Z}/2\mathbb {Z}$ -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 $\mathcal {H}_{g, n}$ . Using this graph complex, we give a sum-over-graphs formula for the $S_n$ -equivariant weight zero compactly supported Euler characteristic of $\mathcal {H}_{g, n}$ . This formula allows for the computer-aided calculation, for each $g\le 7$ , of the generating function $\mathsf {h}_g$ 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 $\Delta $ -complex. We use these complexes to generalize our formula for $\mathsf {h}_g$ to moduli spaces of n-pointed smooth abelian covers of genus zero curves.

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Section: Related Work On the Cohomology Of H 𝑔𝑛mentioning

confidence: 97%