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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Automorphisms of curves and Weierstrass semigroups for Harbater-Katz-Gabber covers

Authors: Sotiris Karanikolopoulos and Aristides Kontogeorgis
Journal: Trans. Amer. Math. Soc. 371 (2019), 6377-6402
MSC (2010): Primary 14H37, 14H55, 11G20; Secondary 20M14, 14H10
Published electronically: February 1, 2019
MathSciNet review: 3937329
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Abstract: We study $ p$-group Galois covers $ X \rightarrow \mathbb{P}^1$ with only one fully ramified point in characteristic $ p>0$. These covers are important because of the Harbater-Katz-Gabber compactification theorem of Galois actions on complete local rings. The sequence of ramification jumps is related to the Weierstrass semigroup of the global cover at the stabilized point. We determine explicitly the jumps of the ramification filtrations in terms of pole numbers. We give applications for curves with zero $ p$-rank: we focus on curves that admit a big action. Moreover, we initiate the study of the Galois module structure of polydifferentials.

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Additional Information

Sotiris Karanikolopoulos
Affiliation: Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany

Aristides Kontogeorgis
Affiliation: Department of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis, 15784 Athens, Greece

Keywords: Automorphisms, curves, numerical semigroups, Harbater--Katz--Gabber covers, zero $p$--rank, big actions, Galois module structure
Received by editor(s): September 24, 2015
Received by editor(s) in revised form: December 22, 2016, July 27, 2017, and November 27, 2017
Published electronically: February 1, 2019
Additional Notes: The first author was supported by a Dahlem Research School and Marie Curie Cofund fellowship; he is also a member of the SFB 647 project Space–Time–Matter, Analytic and Geometric Structures.
The second author was supported by the European Union (European Social Fund—ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF)—Research Funding Program: THALIS
Article copyright: © Copyright 2019 American Mathematical Society