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A class of non-rational surface singularities with bijective Nash map

Camille Plénat, Patrick Popescu-Pampu (2006)

Bulletin de la Société Mathématique de France

Let be a germ of complex analytic normal surface. On its minimal resolution, we consider the reduced exceptional divisor and its irreducible components , . The Nash map associates to each irreducible component of the space of arcs through on the unique component of cut by the strict transform of the generic arc in . Nash proved its injectivity and asked if it was bijective. As a particular case of our main theorem, we prove that this is the case if for any .

A combinatorial approach to singularities of normal surfaces

Sandro Manfredini (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper we study generic coverings of branched over a curve s.t. the total space is a normal analytic surface, in terms of a graph representing the monodromy of the covering, called monodromy graph. A complete description of the monodromy graphs and of the local fundamental groups is found in case the branch curve is (with ) and the degree of the cover is equal to or .

A KAM phenomenon for singular holomorphic vector fields

Laurent Stolovitch (2005)

Publications Mathématiques de l'IHÉS

Let X be a germ of holomorphic vector field at the origin of Cn and vanishing there. We assume that X is a good perturbation of a “nondegenerate” singular completely integrable system. The latter is associated to a family of linear diagonal vector fields which is assumed to have nontrivial polynomial first integrals (they are generated by the so called “resonant monomials”). We show that X admits many invariant analytic subsets in a neighborhood of the origin. These are biholomorphic to the intersection...

A new proof of desingularization over fields of characteristic zero.

Santiago Encinas, Orlando Villamayor (2003)

Revista Matemática Iberoamericana

We present a proof of embedded desingularization for closed subschemes which does not make use of Hilbert-Samuel function and avoids Hironaka's notion of normal flatness (see also [171 page 224). Given a subscheme defined by equations, we prove that embedded desingularization can be achieved by a sequence of monoidal transformations; where the law of transformation on the equations defining the subscheme is simpler then that used in Hironaka 's procedure. This is done by showing that desingularization...

A note on M. Soares’ bounds

Eduardo Esteves, Israel Vainsencher (2006)

Annales de l’institut Fourier

We give an intersection theoretic proof of M. Soares’ bounds for the Poincaré-Hopf index of an isolated singularity of a foliation of .

A note on projective Levi flats and minimal sets of algebraic foliations

Alcides Lins Neto (1999)

Annales de l'institut Fourier

In this paper we prove that holomorphic codimension one singular foliations on have no non trivial minimal sets. We prove also that for , there is no real analytic Levi flat hypersurface in .

A note on singularities at infinity of complex polynomials

Adam Parusiński (1997)

Banach Center Publications

Let f be a complex polynomial. We relate the behaviour of f “at infinity” to the sheaf of vanishing cycles of the family of projective closures of fibres of f. We show that the absence of such cycles: (i) is equivalent to a condition on the asymptotic behaviour of gradient of f known as Malgrange’s Condition, (ii) implies the -triviality of f. If the support of sheaf of vanishing cycles of is a finite set, then it detects precisely the change of the topology of the fibres of f. Moreover, in...

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