On Innermost Circles of the Sets of Singular Values for Generic Deformations of Isolated Singularities
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On Innermost Circles of the Sets of Singular Values for Generic Deformations of Isolated Singularities Kazumasa Inaba1 · Masaharu Ishikawa1 · Masayuki Kawashima2 · Nguyen Tat Thang3
Received: 23 June 2016 / Revised: 27 October 2016 / Accepted: 6 November 2016 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2017
Abstract We will show that for each k = 1, there exists an isolated singularity of a real analytic map from R4 to R2 which admits a real analytic deformation such that the set of singular values of the deformed map has a simple, innermost component with k outward cusps and no inward cusps. Conversely, such a singularity does not exist if k = 1. Keywords Stable map · Excellent map · Critical value · Higher differential · Mixed polynomial Mathematics Subject Classification (2010) Primary 57R45 · Secondary 58C27 · 14B05
Masaharu Ishikawa
[email protected] Kazumasa Inaba [email protected] Masayuki Kawashima [email protected] Nguyen Tat Thang [email protected] 1
Mathematical Institute, Tohoku University, Sendai, 980-8578, Japan
2
Department of Information Science, Okayama University of Science, 1-1 Ridai-cho, Kitaku, Okayama 700-0005, Japan
3
Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Cau Giay, 10307 Hanoi, Vietnam
K. Inaba et al.
1 Introduction In this paper, we study what kinds of curves can appear as the sets of singular values of generic maps obtained by deforming isolated singularities of real analytic maps from R4 to R2 . Throughout the paper, all deformations are real analytic otherwise mentioned. A smooth map is said to be excellent if it has only fold and cusp singularities. In [4], the authors proved that a real deformation of a Brieskorn singularity defined by adding complex-conjugate linear terms is an excellent map in general and estimated the number of cusps appearing after the deformation. During the research, we could see many kinds of curves as the set of singular values obtained after deformations. To find a characterization of such curves, as a basic observation, we focus on an innermost component of the set of singular values and study it under the assumption that the innermost component is a simple closed curve with only outward cusps. To state our result, we first give a precise definition of an innermost circle and also give definitions of outward and inward cusps. Let C be a union of closed curves in R2 which are immersed except a finite number of cusps. A component C0 of C is said to be innermost if there exists an open disk D in R2 containing C0 such that D does not intersect the other components of C. Suppose that C0 is simple. Then, a cusp in C0 is said to be outward if the disk with cusps bounded by C0 is locally convex near the cusp. Otherwise it is said to be inward. For a real analytic map f : R4 → R2 , let S(f ) denote the set of singular points of f . The main theorem is the following. Theorem 1.1 For each k = 1, there exists
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