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Orbifold Jacobian algebras for exceptional unimodal singularities. (English) Zbl 1393.14003

Let \(f\) be an invertible polynomial defining an exceptional unimodal singularity from the list of V. I. Arnol’d [Russ. Math. Surv. 30, No. 5, 1–75 (1975; Zbl 0343.58001)]. The authors show that the orbifold Jacobian algebra of \(f\) (see [A. Basalaev et al., “Orbifold Jacobian algebras for invertible polynomials”, Preprint, arXiv:1608.08962]) is isomorphic to the ordinary Jacobian algebra of the Berglund-Hübsch transpose of another invertible polynomial (see [P. Berglund and T. Hübsch, Nucl. Phys., B 393, No. 1–2, 377–391 (1993; Zbl 1245.14039)]), which determines the strange dual singularity in the sense of Arnold (cf. also [W. Ebeling and A. Takahashi, Arnold Math. J. 2, No. 3, 277–298 (2016; Zbl 1357.14054)]).

MSC:

14B05 Singularities in algebraic geometry
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
57R45 Singularities of differentiable mappings in differential topology
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References:

[1] Arnold, V.I., Gusein-Zade, S., Varchenko, A.: Singularities of Differentiable Maps, vol. 2. Birkhäuser, Boston (2012) · Zbl 1290.58001 · doi:10.1007/978-0-8176-8340-5
[2] Basalaev, A., Takahashi, A., Werner, E.: Orbifold Jacobian algebras for invertible polynomials. arXiv:1608.08962 · Zbl 1393.14003
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