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On Halphen’s Theorem and some generalizations

Alcides Lins Neto (2006)

Annales de l’institut Fourier

Let M n be a germ at 0 ∈ ℂ m of an irreducible analytic set of dimension n , where n ≥ 2 and 0 is a singular point of M . We study the question: when does there exist a germ of holomorphic map φ : ( ℂ n , 0 ) → ( M , 0 ) such that φ - 1 ( 0 ) = { 0 } ? We prove essentialy three results. In Theorem 1 we consider the case where M is a quasi-homogeneous complete intersection of k polynomials F = ( F 1 , ... , F k ) , that is there exists a linear holomorphic vector field X on ℂ m , with eigenvalues λ 1 , ... , λ m ∈ ℚ + such that X ( F T ) = U · F T , where U is a k × k matrix with entries in 𝒪 m . We prove that if there exists...

On higher dimensional Hirzebruch-Jung singularities.

Patrick Popescu-Pampu (2005)

Revista Matemática Complutense

A germ of normal complex analytical surface is called a Hirzebruch-Jung singularity if it is analytically isomorphic to the germ at the 0-dimensional orbit of an affine toric surface. Two such germs are known to be isomorphic if and only if the toric surfaces corresponding to them are equivariantly isomorphic. We extend this result to higher-dimensional Hirzebruch-Jung singularities, which we define to be the germs analytically isomorphic to the germ at the 0-dimensional orbit of an affine toric...

On homological index

Bahman Khanedani (2000)

Annales de la Faculté des sciences de Toulouse : Mathématiques

On meromorphic functions defined by a differential system of order 1

Tristan Torrelli (2004)

Bulletin de la Société Mathématique de France

Given a germ h of holomorphic function on ( ℂ n , 0 ) , we study the condition: “the ideal Ann 𝒟 1 / h is generated by operators of order1”. We obtain here full characterizations in the particular cases of Koszul-free germs and unreduced germs of plane curves. Moreover, we prove that this condition holds for a special type of hyperplane arrangements. These results allow us to link this condition to the comparison of de Rham complexes associated with h .

On second order Thom-Boardman singularities

László M. Fehér, Balázs Kőműves (2006)

Fundamenta Mathematicae

We derive closed formulas for the Thom polynomials of two families of second order Thom-Boardman singularities Σ i , j . The formulas are given as linear combinations of Schur polynomials, and all coefficients are nonnegative.

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