The local geometry of the stack of $A_r$-stable curves
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918420568604672 |
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| author | Gori, Davide Modin, Ludvig Pernice, Michele |
| author_facet | Gori, Davide Modin, Ludvig Pernice, Michele |
| contents | In this paper we study the local geometry of the stack of pointed $A_r$-stable curves. In particular, we analyze the deformation theory of $A_r$-stable curves and their automorphism groups in order to study the combinatorics of families of curves over $[\mathbb{A}^1/\mathbb{G}_m]$, and use this to classify all closed points of the stack of $A_r$-stable curves. As a byproduct, we also classify all open substacks of the moduli stack of degree $2$ cyclic covers of $\mathbb{P}^1$ that admit a separated good moduli space. This is the first in a series of three papers aimed at studying obstructions for the existence of good moduli spaces for stacks of curves with $A$-type singularities, and using these to find an open substack of the stack of $A_r$-stable curves that admits a proper non-projective good moduli space when $r=5$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_29853 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The local geometry of the stack of $A_r$-stable curves Gori, Davide Modin, Ludvig Pernice, Michele Algebraic Geometry 14B07, 14D23, 14H10 In this paper we study the local geometry of the stack of pointed $A_r$-stable curves. In particular, we analyze the deformation theory of $A_r$-stable curves and their automorphism groups in order to study the combinatorics of families of curves over $[\mathbb{A}^1/\mathbb{G}_m]$, and use this to classify all closed points of the stack of $A_r$-stable curves. As a byproduct, we also classify all open substacks of the moduli stack of degree $2$ cyclic covers of $\mathbb{P}^1$ that admit a separated good moduli space. This is the first in a series of three papers aimed at studying obstructions for the existence of good moduli spaces for stacks of curves with $A$-type singularities, and using these to find an open substack of the stack of $A_r$-stable curves that admits a proper non-projective good moduli space when $r=5$. |
| title | The local geometry of the stack of $A_r$-stable curves |
| topic | Algebraic Geometry 14B07, 14D23, 14H10 |
| url | https://arxiv.org/abs/2603.29853 |