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Moderately ramified actions in positive characteristic

  • Dino LorenziniEmail author
  • Stefan Schröer
Article

Abstract

In characteristic 2 and dimension 2, wild \({\mathbb Z}/2{{\mathbb {Z}}}\)-actions on k[[uv]] ramified precisely at the origin were classified by Artin, who showed in particular that they induce hypersurface singularities. We introduce in this article a new class of wild quotient singularities in any characteristic \(p>0\) and dimension \(n\ge 2\) arising from certain non-linear actions of \({\mathbb {Z}}/p{\mathbb {Z}}\) on the formal power series ring \(k[[u_1,\dots ,u_n]]\). These actions are ramified precisely at the origin, and their rings of invariants in dimension 2 are hypersurface singularities, with an equation of a form similar to the form found by Artin when \(p=2\). In higher dimension, the rings of invariants are not local complete intersection in general, but remain quasi-Gorenstein. We establish several structure results for such actions and their corresponding rings of invariants.

Mathematics Subject Classification

14B05 14J17 14L15 14E22 13B02 

Notes

Acknowledgements

The authors thank the referee for a careful reading of the manuscript and for useful comments. The authors gratefully acknowledges funding support from the research training group Algebra, Algebraic Geometry, and Number Theory at the University of Georgia, from the National Science Foundation RTG grant DMS-1344994, from the Simons Collaboration Grant 245522, and the research training group GRK 2240: Algebro-geometric Methods in Algebra, Arithmetic and Topology of the Deutsche Forschungsgemeinschaft.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GeorgiaAthensUSA
  2. 2.Mathematisches InstitutHeinrich-Heine-UniversitätDüsseldorfGermany

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