Equisingular deformations of plane curves in arbitrary characteristic
| Autor: | Antonio Campillo, Christoph Lossen, Gert-Martin Greuel |
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| Rok vydání: | 2007 |
| Předmět: |
Pure mathematics
14B05 Algebra and Number Theory 14H20 14B07 Plane curve Multiplicity (mathematics) Mathematics - Commutative Algebra Commutative Algebra (math.AC) Mathematical proof Milnor number Base change Mathematics - Algebraic Geometry Mathematics::Algebraic Geometry Singularity FOS: Mathematics Gravitational singularity Subfunctor Algebraic Geometry (math.AG) Mathematics |
| Zdroj: | Compositio Mathematica. 143:829-882 |
| ISSN: | 1570-5846 0010-437X |
| DOI: | 10.1112/s0010437x07002953 |
| Popis: | In this paper we develop the theory of equisingular deformations of plane curve singularities in arbitrary characteristic. We study equisingular deformations of the parametrization and of the equation and show that the base space of its semiuniveral deformation is smooth in both cases. Our approach through deformations of the parametrization is elementary and we show that equisingular deformations of the parametrization form a linear subfunctor of all deformations of the parametrization. This gives additional information, even in characteristic zero, the case which was treated by J. Wahl. The methods and proofs extend easily to good characteristic, that is, when the characteristic does not divide the multiplicity of any branch of the singularity. In bad characteristic, however, new phenomena occur and we are naturally led to consider weakly trivial respectively weakly equisingular deformations, that is, those which become trivial respectively equisingular after a finite and dominant base change. The semiuniversal base space for weakly equisingular deformations is, in general, not smooth but becomes smooth after a finite and purely inseparable base extension. For the proof of this fact we introduce some constructions which may have further applications in the theory of singularities in positive characteristic. Comment: 56 pages |
| Databáze: | OpenAIRE |
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