Stabilization indices of potentially Mumford curves
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866910280359870464 |
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| author | Obus, Andrew Turchetti, Daniele |
| author_facet | Obus, Andrew Turchetti, Daniele |
| contents | Let $X$ be a smooth projective curve over a complete discretely valued field $K$. Let $L/K$ be the minimal extension such that $X \times_K L$ has a semi-stable model, and write $e(L/K)$ for the ramification index of $L/K$. Let $e(X)$ be the so-called ``stabilization index'' of $X$, defined by Halle and Nicaise as the lcm of the multiplicities of the ``principal'' irreducible components of a minimal regular snc-model of $X$.
It is known that if $L/K$ is tame, then $e(X) = e(L/K)$. If one drops the tameness assumption, but instead assumes that $X$ has index one and potentially multiplicative reduction, Halle and Nicaise ask if the equality $e(X) = e(L/K)$ still holds. We prove that $e(X)$ divides $e(L/K)$ in this situation, but we give examples, in every residue characteristic, of $X$ with $K$-rational points and potentially multiplicative reduction such that $e(X) \neq e(L/K)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_14728 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Stabilization indices of potentially Mumford curves Obus, Andrew Turchetti, Daniele Algebraic Geometry Primary: 11G20, 14B05, Secondary: 14H25, 14J17 Let $X$ be a smooth projective curve over a complete discretely valued field $K$. Let $L/K$ be the minimal extension such that $X \times_K L$ has a semi-stable model, and write $e(L/K)$ for the ramification index of $L/K$. Let $e(X)$ be the so-called ``stabilization index'' of $X$, defined by Halle and Nicaise as the lcm of the multiplicities of the ``principal'' irreducible components of a minimal regular snc-model of $X$. It is known that if $L/K$ is tame, then $e(X) = e(L/K)$. If one drops the tameness assumption, but instead assumes that $X$ has index one and potentially multiplicative reduction, Halle and Nicaise ask if the equality $e(X) = e(L/K)$ still holds. We prove that $e(X)$ divides $e(L/K)$ in this situation, but we give examples, in every residue characteristic, of $X$ with $K$-rational points and potentially multiplicative reduction such that $e(X) \neq e(L/K)$. |
| title | Stabilization indices of potentially Mumford curves |
| topic | Algebraic Geometry Primary: 11G20, 14B05, Secondary: 14H25, 14J17 |
| url | https://arxiv.org/abs/2112.14728 |