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PreprintCC BY 4.0arXiv

Equivalence of Curve Singularities and delta-Invariants

  • Reinhold Hübl
  • Irena Swanson
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Abstract

We prove that if two parameterizations of a complete reduced noetherian curve over an algebraically closed field agree modulo a sufficiently large power of the maximal ideal, then the two parameterizations are equivalent. This strengthens some bounds from Greuel and Pfister. In addition, we prove that if two reduced and irreducible curve singularities are isomorphic modulo sufficiently high (and identical) powers of their respective maximal ideals, then the completions of the two curves are isomorphic, and the isomorphism of the full completions agrees with the original isomorphism modulo some lower power of the maximal ideals. We provide a new and better bound on the lower power, strengthening the bound in Hironaka

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Cite

BibTeXRIS
BibTeX
@misc{hubl2026equivalence,
  title = {{Equivalence of Curve Singularities and delta-Invariants}},
  author = {Reinhold Hübl and Irena Swanson},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2608.18344},
  eprint = {2608.18344},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.18344},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Equivalence of Curve Singularities and delta-Invariants
AU  - Reinhold Hübl
AU  - Irena Swanson
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2608.18344
UR  - https://arxiv.org/abs/2608.18344
AB  - We prove that if two parameterizations of a complete reduced noetherian curve over an algebraically closed field agree modulo a sufficiently large power of the maximal ideal, then the two parameterizations are equivalent. This strengthens some bounds from Greuel and Pfister. In addition, we prove that if two reduced and irreducible curve singularities are isomorphic modulo sufficiently high (and identical) powers of their respective maximal ideals, then the completions of the two curves are isomorphic, and the isomorphism of the full completions agrees with the original isomorphism modulo some lower power of the maximal ideals. We provide a new and better bound on the lower power, strengthening the bound in Hironaka
ER  -