A Few Problems on Monodromy and Discriminants

Vassiliev, Victor A.

doi:10.1007/s40598-015-0011-9

Cited by 6 publications

(2 citation statements)

References 14 publications

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“…V. A. Vassiliev listed some problems about the Lyashko-Looijenga mapping for non-simple singularities in [Va15].…”

Section: Special Singularitiesmentioning

confidence: 99%

Distinguished bases and monodromy of complex hypersurface singularities

Ebeling¹

2019

Preprint

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We give a survey on some aspects of the topological investigation of isolated singularities of complex hypersurfaces by means of Picard-Lefschetz theory. We focus on the concept of distinguished bases of vanishing cycles and the concept of monodromy. ContentsIntroduction 1 1. Distinguished bases of vanishing cycles 3 2. Coxeter-Dynkin diagram and Seifert form 11 3. Change of basis 13 4. Computation of intersection matrices 18 5. The discriminant and the Lyashko-Looijenga map 21 6. Special singularities 24 7. The monodromy group 31 8. Topological equivalence 35 References 37 Index 46

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“…V. A. Vassiliev listed some problems about the Lyashko-Looijenga mapping for non-simple singularities in [Va15].…”

Section: Special Singularitiesmentioning

confidence: 99%

Distinguished bases and monodromy of complex hypersurface singularities

Ebeling¹

2019

Preprint

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“…Here, we relate the study of amoebas of curves to the Lyashko–Looijenga map. The latter is a fundamental tool in singularity theory and the study of Hurwitz spaces (see, for example, ). In particular, the ability to extend the map

ℓ ℓ

to nodal curves has powerful applications in the latter topics.…”

Section: Introductionmentioning

confidence: 99%

Amoebas of curves and the Lyashko–Looijenga map

Lang

2019

J. London Math. Soc.

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For any curve V in a toric surface X, we study the critical locus S ⊂ V of the moment map μ from V to its compactified amoeba μ(V). For any complete linear system |L| given by an ample line bundle L on X, we show that the critical locus S ⊂ V is smooth as long as the curve V is outside of a subset of real codimension 1 in |L|. In particular, the complement of the latter subset appears to be disconnected for general L. It suggests a classification problem analogous to Hilbert's Sixteenth Problem, namely the topological classification of pairs (V, S) for curves V ∈ |L|. The description of the critical locus S in terms of the logarithmic Gauß map γ : V → CP 1 relates the latter problem to the study of the Lyashko-Looijenga map ( ). The map associates to a generic curve V ∈ |L| the unordered set of the critical values of γ on CP 1 . We prove two statements concerning that are crucial for our classification problem: the map is algebraic; the map extends to nodal curves in |L|. This fact allows us to construct many examples of pairs (V, S) by perturbing nodal curves. Contents

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Distinguished Bases and Monodromy of Complex Hypersurface Singularities

Ebeling

2020

Handbook of Geometry and Topology of Singularities I

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A Few Problems on Monodromy and Discriminants

Cited by 6 publications

References 14 publications

Distinguished bases and monodromy of complex hypersurface singularities

Distinguished bases and monodromy of complex hypersurface singularities

Amoebas of curves and the Lyashko–Looijenga map

Distinguished Bases and Monodromy of Complex Hypersurface Singularities

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