Geodesic
On the deformation of the exceptional unimodal singularities
Journal of Singularities, Tome 23 (2021), pp. 1-14

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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.
DOI : 10.5427/jsing.2021.23a
Classification : 32S25, 58K60, 58K05 
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     title = {On the deformation of the exceptional unimodal singularities},
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     doi = {10.5427/jsing.2021.23a},
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Naohiko Kasuya; Atsuhide Mori. On the deformation of the exceptional unimodal singularities. Journal of Singularities, Tome 23 (2021), pp. 1-14. doi: 10.5427/jsing.2021.23a

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