Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 4

Special Issue: In Honor of Prof. Gert-Martin Greuel’s 75th Birthday

Guest Editors: Igor Burban, Stanislaw Janeczko, Gerhard Pfister, Stephen S.T. Yau, and Huaiqing Zuo

Parametrization simple irreducible curve singularities in arbitrary characteristic

Pages: 1053 – 1066

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n4.a6


Hong Duc Nguyen (Basque Center for Applied Mathematics, Bilbao, Bizkaia, Spain; and TIMAS, Thang Long University, Nghiem Xuan Yem, Hanoi, Vietnam)


We study the classification of plane curve singularities in arbitrary characteristic. We first give a bound for the determinacy of a plane curve singularity with respect to pararametrization equivalence in terms of its conductor. We then define the notion of parametrization modality for plane curve singularities. Finally, we re-visit Mehmood–Pfister’s classification of parametrization simple plane curve singularities which are irreducible by giving a concrete list of normal forms of equations and parametrizations. In characteristic zero, the classification of parametrization simple irreducible plane curve singularities was achieved by Bruce and Gaffney.

The author’s research is supported by Juan de la Cierva Incorporación IJCI-2016-29891, the ERCEA Consolidator Grant 615655 NMST, and the National Foundation for Science and Technology Development (NAFOSTED), Grant number 101.04-2019.316, Vietnam.

Received 2 January 2019

Accepted 11 December 2019

Published 13 November 2020