## Journal of the Mathematical Society of Japan

### A classification of weighted homogeneous Saito free divisors

Jiro SEKIGUCHI

#### Abstract

We describe an approach to classification of weighted homogeneous Saito free divisors in $\mbi{C}^{3}$. This approach is mainly based on properties of Lie algebras of vector fields tangent to reduced hypersurfaces at their non-singular points. In fact we also obtain a classification of such Lie algebras having similar properties as ones for discriminants associated with irreducible real reflection groups of rank 3. Among other things we briefly discuss some applications to the theory of discriminants of irreducible reflection groups of rank 3, some interesting relationships with root systems of types $E_{6}$, $E_{7}$, $E_{8}$, and few examples in higher dimensional cases.

#### Article information

Source
J. Math. Soc. Japan Volume 61, Number 4 (2009), 1071-1095.

Dates
First available in Project Euclid: 6 November 2009

http://projecteuclid.org/euclid.jmsj/1257520500

Digital Object Identifier
doi:10.2969/jmsj/06141071

Zentralblatt MATH identifier
05651144

Mathematical Reviews number (MathSciNet)
MR2588504

#### Citation

SEKIGUCHI, Jiro. A classification of weighted homogeneous Saito free divisors. Journal of the Mathematical Society of Japan 61 (2009), no. 4, 1071--1095. doi:10.2969/jmsj/06141071. http://projecteuclid.org/euclid.jmsj/1257520500.

#### References

• A. G. Aleksandrov, Milnor numbers of nonisolated Saito singularities, Funct. Anal. Appl., 21 (1987), 1–9.
• A. G. Aleksandrov, Nonisolated hypersurfaces singularities, Theory of singularities and its applications, Adv. Soviet Math., 1 (1990), 211–246.
• A. G. Aleksandrov, Moduli of logarithmic connections along free divisor, Topology and geometry : commemorating SISTAG, Contemp. Math., 314 (2002), 1–23.
• A. G. Aleksandrov, Nonisolated Saito singularities, Mat. Sb. (N.S.), 137(179) (1988), 554–567, 576 (Russian); translation in Math. USSR-Sb., 65 (1990), 561–574.
• A. G. Aleksandrov and J. Sekiguchi, Free deformations of hypersurface singularities, to appear in RIMS Kokyuroku.
• V. I. Arnol'd, S. M. Guseĭn-Zade and A. N. Varchenko, Singularities of differentiable maps, Vol. I., The classification of critical points, caustics and wave fronts, Monographs in Mathematics, 82, Birkhäuser Boston, Inc., Boston, MA, 1985.
• E. Brieskorn and K. Saito, Artin-Gruppen und Coxeter-Gruppen, Invent. Math., 17 (1972), 245–271.
• P. Cartier, Les arrangements d'hyperplans: un chapitre de geometrie combinatoire, Semin. Bourbaki, 33e annee, Vol. 1980/81, Exp. No. 561, Lecture Notes in Math., 901, Springer-Verlag, 1981, pp. 1–22.
• J. Damon, On the freeness of equisingular deformations of plane curve singularities, Topology Appl., 118 (2002), 31–43.
• P. Deligne, Les immeubles des groupes de tresses généralisés, Invent. Math., 17 (1972), 273–302.
• M. Granger, D. Mond, A. N. Reyes and M. Schulze, Linear free divisors, preprint.
• T. Ishibe, Master thesis presented to RIMS, Kyoto University, 2007.
• P. Orlik and H. Terao, Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften, Fundamental Principles of Mathematical Sciences, 300, Springer-Verlag, Berlin, 1992.
• K. Saito, On the uniformization of complements of discriminant loci, RIMS Kokyuroku, 287 (1977), 117–137.
• K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci., Univ. Tokyo Sect. IA Math., 27 (1980), 265–291.
• K. Saito, On a linear structure of the quotient variety by a finite reflection group, Publ. Res. Inst. Math. Sci. Kyoto Univ., 29 (1993), 535–579.
• J. Sekiguchi, Some topics related with discriminant polynomials, RIMS Kokyuroku, 810 (1992), 85–94.
• J. Sekiguchi, Three dimensional Saito free divisors and deformations of singular curves, J. Siberian Federal Univ., Mathematics & Physics, 1 (2008), 33–41.
• P. Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Math., Springer 815, Springer-Verlag, 1980.
• H. Terao, Arrangements of hyperplanes and their freeness, I, II, J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 27 (1980), 293–320.
• T. Yano and J. Sekiguchi, The microlocal structure of weighted homogenous polynomials associated with Coxeter systems, I, Tokyo J. Math., 2 (1979), 193–219.
• T. Yano and J. Sekiguchi, The microlocal structure of weighted homogenous polynomials associated with Coxeter systems, II, Tokyo J. Math., 4 (1981), 1–34.
• T. Yano and J. Sekiguchi, The microlocal structure of weighted homogenous polynomials associated with Coxeter systems (with Appendix on $GL(2)$), RIMS Kokyuroku, 281 (1976), 40–105.