Explicit resolution of weak wild quotient singularities on arithmetic surfaces
Authors:
Andrew Obus and Stefan Wewers
Journal:
J. Algebraic Geom. 29 (2020), 691-728
DOI:
https://doi.org/10.1090/jag/745
Published electronically:
December 6, 2019
MathSciNet review:
4158463
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Abstract |
References |
Additional Information
Abstract: A weak wild arithmetic quotient singularity arises from the quotient of a smooth arithmetic surface by a finite group action, where the inertia group of a point on a closed characteristic $p$ fiber is a $p$-group acting with smallest possible ramification jump. In this paper, we give complete explicit resolutions of these singularities using deformation theory and valuation theory, taking a more local perspective than previous work has taken. Our descriptions answer several questions of Lorenzini. Along the way, we give a valuation-theoretic criterion for a normal snc-model of $\mathbb {P}^1$ over a discretely valued field to be regular.
References
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References
- M. Artin, Wildly ramified $Z/2$ actions in dimension two, Proc. Amer. Math. Soc. 52 (1975), 60–64. MR 374136, DOI https://doi.org/10.2307/2040100
- José Bertin and Ariane Mézard, Déformations formelles des revêtements sauvagement ramifiés de courbes algébriques, Invent. Math. 141 (2000), no. 1, 195–238 (French, with English summary). MR 1767273, DOI https://doi.org/10.1007/s002220000071
- Egbert Brieskorn, Rationale Singularitäten komplexer Flächen, Invent. Math. 4 (1967/1968), 336–358 (German). MR 0222084, DOI https://doi.org/10.1007/BF01425318
- Brian Conrad, Bas Edixhoven, and William Stein, $J_1(p)$ has connected fibers, Doc. Math. 8 (2003), 331–408. MR 2029169
- Gunther Cornelissen and Fumiharu Kato, Equivariant deformation of Mumford curves and of ordinary curves in positive characteristic, Duke Math. J. 116 (2003), no. 3, 431–470. MR 1958094, DOI https://doi.org/10.1215/S0012-7094-03-11632-4
- Anne Frühbis-Krüger and Stefan Wewers, Desingularization of arithmetic surfaces: algorithmic aspects, Algorithmic and experimental methods in algebra, geometry, and number theory, Springer, Cham, 2017, pp. 231–252. MR 3792728
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
- David Harbater, Moduli of $p$-covers of curves, Comm. Algebra 8 (1980), no. 12, 1095–1122. MR 579791, DOI https://doi.org/10.1080/00927878008822511
- Hiroyuki Ito and Stefan Schröer, Wild quotient surface singularities whose dual graphs are not star-shaped, Asian J. Math. 19 (2015), no. 5, 951–986. MR 3431685, DOI https://doi.org/10.4310/AJM.2015.v19.n5.a7
- William B. Jones and Wolfgang J. Thron, Continued fractions, Analytic theory and applications; with a foreword by Felix E. Browder, with an introduction by Peter Henrici, Encyclopedia of Mathematics and its Applications, vol. 11, Addison-Wesley Publishing Co., Reading, Mass., 1980. MR 595864
- Joseph Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes Études Sci. Publ. Math. 36 (1969), 195–279. MR 0276239
- Joseph Lipman, Desingularization of two-dimensional schemes, Ann. Math. (2) 107 (1978), no. 1, 151–207. MR 0491722, DOI https://doi.org/10.2307/1971141
- Qing Liu, Algebraic geometry and arithmetic curves, translated from the French by Reinie Erné; Oxford Science Publications, Oxford Graduate Texts in Mathematics, vol. 6, Oxford University Press, Oxford, 2002. MR 1917232
- Dino Lorenzini, Wild quotient singularities of surfaces, Math. Z. 275 (2013), no. 1-2, 211–232. MR 3101805, DOI https://doi.org/10.1007/s00209-012-1132-7
- Dino Lorenzini, Wild models of curves, Algebra Number Theory 8 (2014), no. 2, 331–367. MR 3212859, DOI https://doi.org/10.2140/ant.2014.8.331
- Dino Lorenzini, Wild quotients of products of curves, Eur. J. Math. 4 (2018), no. 2, 525–554. MR 3799154, DOI https://doi.org/10.1007/s40879-017-0174-0
- Saunders Mac Lane, A construction for absolute values in polynomial rings, Trans. Amer. Math. Soc. 40 (1936), no. 3, 363–395. MR 1501879, DOI https://doi.org/10.2307/1989629
- Andrew Obus, Conductors of wild extensions of local fields, especially in mixed characteristic $(0,2)$, Proc. Amer. Math. Soc. 142 (2014), no. 5, 1485–1495. MR 3168456, DOI https://doi.org/10.1090/S0002-9939-2014-11881-8
- Julian Rüth, Models of curves and valuations, PhD thesis, available at http://dx.doi.org/10.18725/OPARU-3275, 2014.
- Jean-Pierre Serre, Corps locaux, Deuxième édition, Publications de l’Université de Nancago, No. VIII, Hermann, Paris, 1968 (French). MR 0354618
- Jean-Pierre Serre, Galois cohomology, translated from the French by Patrick Ion and revised by the author, Springer-Verlag, Berlin, 1997. MR 1466966
- Christian Steck, Resolution of tame cyclic quotient singularities on fibered surfaces, PhD thesis, Ulm University, 2018.
Additional Information
Andrew Obus
Affiliation:
Baruch College, 1 Bernard Baruch Way, New York, New York 10010
MR Author ID:
890287
ORCID:
0000-0003-2358-4726
Email:
andrewobus@gmail.com
Stefan Wewers
Affiliation:
Universität Ulm, Helmholzstraße 18, 89081 Ulm, Germany
MR Author ID:
652833
Email:
stefan.wewers@uni-ulm.de
Received by editor(s):
June 10, 2018
Received by editor(s) in revised form:
March 5, 2019, and May 5, 2019
Published electronically:
December 6, 2019
Additional Notes:
The first author was supported by NSF Grant DMS-1602054
Article copyright:
© Copyright 2019
University Press, Inc.