Mond's conjecture for maps between curves

Ament, D. A. H.; Nuño–Ballesteros, J. J.

doi:10.1002/mana.201600483

Mathematische Nachrichten

2017

DOI: 10.1002/mana.201600483

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Mond's conjecture for maps between curves

D. A. H. Ament

J. J. Nuño–Ballesteros

Abstract: A theorem by D. Mond shows that if ∶ (ℂ, 0) → ( ℂ 2 , 0 ) is finite and has has degree one onto its image ( , 0), then the A -codimension is less than or equal to the image Milnor number ( ), with equality if and only if ( , 0) is weighted homogeneous.Here we generalize this result to the case of a map germ ∶ ( , 0), where ( , 0) is a plane curve singularity. K E Y W O R D SA -codimension, image Milnor number, curve singularities M S C ( 2 0 1 0 ) Primary: 32S30; Secondary: 32S05, 58K60

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Cited by 3 publications

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Disentangling mappings defined on ICIS

Fernández-Hernández,

Nuño-Ballesteros

2024

Rev Mat Complut

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We study germs of hypersurfaces $$(Y,0)\subset (\mathbb {C}^{n+1},0)$$ ( Y , 0 ) ⊂ ( C n + 1 , 0 ) that can be described as the image of $${\mathscr {A}}$$ A -finite mappings $$f:(X,S)\rightarrow (\mathbb {C}^{n+1},0)$$ f : ( X , S ) → ( C n + 1 , 0 ) defined on an icis (X, S) of dimension n. We extend the definition of the Jacobian module given by Fernández de Bobadilla, Nuño-Ballesteros and Peñafort-Sanchis when $$X=\mathbb {C}^n$$ X = C n , which controls the image Milnor number $$\mu _I(X,f)$$ μ I ( X , f ) . We apply these results to prove the case $$n=2$$ n = 2 of the generalised Mond conjecture, which states that $${\mu _I(X,f)\ge \text{ codim}_{\mathscr {A}_e}(X,f)}$$ μ I ( X , f ) ≥ codim A e ( X , f ) , with equality if (Y, 0) is weighted homogeneous.

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Disentangling mappings defined on ICIS

Fernández-Hernández,

Nuño-Ballesteros

2024

Rev Mat Complut

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Singularities of mappings on ICIS and applications to Whitney equisingularity

Conejero

Nuño–Ballesteros²

2022

Advances in Mathematics

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Image Milnor number and 𝒜_e-codimension for maps between weighted homogeneous irreducible curves

Ament

Nuño–Ballesteros

Tomazella

2019

Advances in Geometry

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Abstract Let (X, 0) ⊂ (ℂn, 0) be an irreducible weighted homogeneous singularity curve and let f : (X, 0) → (ℂ2, 0) be a finite map germ, one-to-one and weighted homogeneous with the same weights of (X, 0). We show that 𝒜e-codim(X, f) = μI(f), where the 𝒜e-codimension 𝒜e-codim(X, f) is the minimum number of parameters in a versal deformation and μI(f) is the image Milnor number, i.e. the number of vanishing cycles in the image of a stabilization of f.

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Mond's conjecture for maps between curves

Cited by 3 publications

References 15 publications

Disentangling mappings defined on ICIS

Disentangling mappings defined on ICIS

Singularities of mappings on ICIS and applications to Whitney equisingularity

Image Milnor number and 𝒜_e-codimension for maps between weighted homogeneous irreducible curves

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Mond's conjecture for maps between curves

Cited by 3 publications

References 15 publications

Disentangling mappings defined on ICIS

Disentangling mappings defined on ICIS

Singularities of mappings on ICIS and applications to Whitney equisingularity

Image Milnor number and 𝒜 e -codimension for maps between weighted homogeneous irreducible curves

Contact Info

Product

Resources

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Image Milnor number and 𝒜_e-codimension for maps between weighted homogeneous irreducible curves