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Right simple singularities in positive characteristic

  • Gert-Martin Greuel EMAIL logo und Hong Duc Nguyen
Veröffentlicht/Copyright: 10. Januar 2014
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Journal für die reine und angewandte Mathematik (Crelles Journal)
Aus der Zeitschrift Journal für die reine und angewandte Mathematik (Crelles Journal) Band 2016 Heft 712

Abstract

We classify isolated singularities f∈K[[x1,...,xn]], which are simple, i.e. have no moduli, with respect to right equivalence, where K is an algebraically closed field of characteristic p > 0. For K=ℝ or ℂ this classification was initiated by Arnol'd, resulting in the famous ADE-series. The classification with respect to contact equivalence for p > 0 was done by Greuel and Kröning with a result similar to Arnol'd's. It is surprising that with respect to right equivalence and any given p > 0 we have only finitely many simple singularities, i.e. there are only finitely many k such that Ak and Dk are right simple, all the others have moduli. We conjecture a similar finiteness result for singularities with an arbitrary number of moduli. A major point of this paper is the generalisation of the notion of modality to the algebraic setting, its behaviour under morphisms, and its relations to formal deformation theory. As an application we show that the modality is semicontinuous in any characteristic.

Funding source: DAAD (Germany)

Funding source: NAFOSTED (Vietnam)

Funding source: Mathematisches Forschungsinstitut Oberwolfach

Award Identifier / Grant number: OWLF programme

We would like to thank the referees for their careful reading of the manuscript and helpful comments which improved the presentation of this paper.

Received: 2012-8-3
Revised: 2013-11-28
Published Online: 2014-1-10
Published in Print: 2016-3-1

© 2016 by De Gruyter

Artikel in diesem Heft

  1. Frontmatter
  2. Refined semiclassical asymptotics for fractional powers of the Laplace operator
  3. Projective metric number theory
  4. Pseudo-parabolic regularization of forward-backward parabolic equations: Power-type nonlinearities
  5. Right simple singularities in positive characteristic
  6. Discriminants and Artin conductors
  7. Hilbertian fields and Galois representations
  8. Simple endotrivial modules for quasi-simple groups
  9. A reconstruction theorem for abelian categories of twisted sheaves
  10. Hyperkähler manifolds of Jacobian type
  11. The algebra of essential relations on a finite set
https://doi.org/10.1515/crelle-2013-0118

Artikel in diesem Heft

  1. Frontmatter
  2. Refined semiclassical asymptotics for fractional powers of the Laplace operator
  3. Projective metric number theory
  4. Pseudo-parabolic regularization of forward-backward parabolic equations: Power-type nonlinearities
  5. Right simple singularities in positive characteristic
  6. Discriminants and Artin conductors
  7. Hilbertian fields and Galois representations
  8. Simple endotrivial modules for quasi-simple groups
  9. A reconstruction theorem for abelian categories of twisted sheaves
  10. Hyperkähler manifolds of Jacobian type
  11. The algebra of essential relations on a finite set
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