Hyperelliptic 𝐴ᵣ-stable curves (and their moduli stack)

Pernice, Michele

doi:10.1090/tran/9164

Hyperelliptic $A_r$-stable curves (and their moduli stack)
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by Michele Pernice

Trans. Amer. Math. Soc.

DOI: https://doi.org/10.1090/tran/9164

Published electronically: April 19, 2024

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Abstract:

This paper is the second in a series of four papers aiming to describe the (almost integral) Chow ring of $\overline {\mathcal {M}}_3$, the moduli stack of stable curves of genus $3$. In this paper, we introduce the moduli stack $\widetilde {\mathcal {H}}_g^r$ of hyperelliptic $A_r$-stable curves and generalize the theory of hyperelliptic stable curves to hyperelliptic $A_r$-stable curves. In particular, we prove that $\widetilde {\mathcal {H}}_g^r$ is a smooth algebraic stack that can be described using cyclic covers of twisted curves of genus $0$ and it embeds in $\widetilde {\mathcal M}_g^r$ (the moduli stack of $A_r$-stable curves) as the closure of the moduli stack of smooth hyperelliptic curves.

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