On the deformation of the exceptional unimodal singularities

Naohiko Kasuya and Atsuhide Mori

Journal of Singularities
volume 23 (2021), 1-14

Received: 31 August 2020. Received in revised form: 26 January 2021

DOI: 10.5427/jsing.2021.23a

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Abstract:

Ebeling and Takahashi considered the deformation of an isolated surface singularity f(x,y,z)-txyz (t in C) for any invertible polynomial f in three variables. In particular, they deformed each of the 14 exceptional unimodal singularities into a cusp singularity. However, their proof is purely algebraic and requires a detailed knowledge of normal forms. In this article, instead of algebraic treatment of the singularity, we observe the critical points of the squared distance function restricted to the singular complex surface in C^3. We show that only one additional critical point emerges via the deformation if and only if f is one of the 14 exceptional unimodal singularities. Moreover, we can determine the approximate location of the critical point when the parameter t is a small positive number. This would be helpful to describe the change of the topology of the complex surface by means of the Morse theory.

2010 Mathematical Subject Classification:

32S25, 58K60, 58K05

Key words and phrases:

exceptional unimodal singularities, cusp singularities, deformations of singularities, critical points, Morse theory

Author(s) information:

Naohiko Kasuya	Atsuhide Mori
Department of Mathematics	Department of Mathematics
Faculty of Science	Osaka Dental University
Kyoto Sangyo University	8-1 Kuzuha Hanazono, Hirakata
Kamigamo Motoyama, Kita-ku	Osaka 573-1121 JAPAN
Kyoto 603-8555 JAPAN
email: nkasuya@cc.kyoto-su.ac.jp	email: mori-a@cc.osaka-dent.ac.jp