## On higher dimensional Hirzebruch-Jung singularities

Venue: | Rev. Mat. Complut |

Citations: | 3 - 1 self |

### BibTeX

@ARTICLE{Popescu-pampu_onhigher,

author = {Patrick Popescu-pampu},

title = {On higher dimensional Hirzebruch-Jung singularities},

journal = {Rev. Mat. Complut},

year = {},

pages = {209--232}

}

### OpenURL

### Abstract

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 variety determined by a lattice and a simplicial cone of maximal dimension. We deduce a normalization algorithm for quasi-ordinary hypersurface singularities. 2000 Mathematics Subject Classification. Primary 32S10; Secondary 14M25. 1

### Citations

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(Show Context)
Citation Context ...oric geometry in the ’70s, Hirzebruch-Jung surface singularities were seen to be precisely the germs analytically isomorphic to the germs of toric surfaces taken at 0-dimensional orbits (see [18] and =-=[10]-=-). It is this view-point which we generalize here. If W is a lattice and σ is a strictly convex finite rational polyhedral cone in WR := W ⊗ R, we denote by M the dual lattice of W and by ˇσ ⊂ MR the ... |

206 |
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(Show Context)
Citation Context ...tion of toric geometry in the ’70s, Hirzebruch-Jung surface singularities were seen to be precisely the germs analytically isomorphic to the germs of toric surfaces taken at 0-dimensional orbits (see =-=[18]-=- and [10]). It is this view-point which we generalize here. If W is a lattice and σ is a strictly convex finite rational polyhedral cone in WR := W ⊗ R, we denote by M the dual lattice of W and by ˇσ ... |

84 |
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Citation Context ...uch that its image is small. Denote d = dim V . Choose a decomposition V = E1 ⊕ E2 ⊕ · · · ⊕ Ed of ρ as a sum of irreducible (1-dimensional) representations. This is possible, since Γ is abelian (see =-=[24]-=-). Denote by E this decomposition. For any g ∈ Γ and any k ∈ {1, ..., d}, g acts on Ek by multiplication by a root of unity e 2iπwk(g) . Here wk(g) ∈ Q is well-defined modulo Z. Define then: wE(g) := ... |

44 |
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(Show Context)
Citation Context ...e action outside the origin, the answer to the question is affirmative. Indeed, in this case the boundary is a generalized lens space and the corresponding result was obtained by Franz (see Dieudonné =-=[7]-=-, page 246). If d ≥ 3, the action ρ(Z) may be non-cyclic, and even if it is cyclic, it may have fixed points. One can decide if Γ(Z) is cyclic by computing the invariant factors of a matrix of present... |

42 |
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Citation Context ...n’s work on rational surface singularities in the ’60s, they were seen to be precisely the rational surface singularities which have as dual resolution graph a segment. This is the definition used in =-=[3]-=-. Hirzebruch-Jung singularities are usually classified up to analytical isomorphism by an odered pair (n, q) ∈ N ∗ × N of coprime numbers with q < n. In order to get this classification, Hirzebruch st... |

41 |
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(Show Context)
Citation Context ...3]uses again Jung’s method in order to prove the existence of a desingularization for complex analytical surfaces which are locally embeddable in C 3 . This last restriction was eliminated by Laufer =-=[15]-=-. An important step in Hirzebruch’s method was to consider the normalizations of the quasi-ordinary germs he arrived at by Jung’s method. He gave explicit constructions of their minimal resolutions by... |

31 |
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Citation Context ... is simply connected, and so the restriction of µ over the smooth part of Z(W, σ) is a universal covering map. The uniqueness in theorem 4.1 implies the proposition. □ Following a terminology used in =-=[6]-=-, we define: Definition 4.3 The morphism µ obtained by Riemann extension of the universal covering map of the smooth part of (Z, 0) is called the orbifold map associated to (Z, 0). Denote by Γ(Z) the ... |

30 |
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(Show Context)
Citation Context ...nt ∆Y (f) of f, which has therefore a dominating exponent. Definition 2.2 Let f ∈ C{X}[Y ] be unitary. If ∆Y (f) has a dominating exponent, we say that f is quasi-ordinary. The following theorem (see =-=[1]-=-, [17]), generalizes the theorem of NewtonPuiseux for plane curves: Theorem 2.3 (Jung-Abhyankar) If f ∈ C{X}[Y ] is quasi-ordinary, then the set R(f) of roots of f embeds canonically in the algebra ˜ ... |

25 |
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Citation Context ...rsection of a representative of (S, 0) with a sufficiently small euclidean sphere centered at 0 in an arbitrary system of local coordinates at 0. It is independent of these choices (Durfee’s proof in =-=[9]-=- for algebraic varieties extends to analytical ones). Hirzebruch [13] noticed that the abstract boundary of a bidimensional Hirzebruch-Jung singularity (Z, 0) of type An,q is a lens space L(n, q). As ... |

21 |
Über vierdimensionale Riemannsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen
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(Show Context)
Citation Context ...5] in order to prove the existence of a resolution of the singularities of a complex algebraic surface. This work is considered by Zariski [26] to be the first rigorous proof of this fact. Hirzebruch =-=[13]-=-uses again Jung’s method in order to prove the existence of a desingularization for complex analytical surfaces which are locally embeddable in C 3 . This last restriction was eliminated by Laufer [1... |

20 | Local classification of quotients of complex manifolds by discontinuous groups - Prill - 1967 |

14 |
Proper holomorphic mappings of complex spaces
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(Show Context)
Citation Context ...point set is a hyperplane of V . A finite group Γ ⊂ GL(V ) is called small (see Prill [22]) if it contains no complex reflections. Let us recall a generalization of the Riemann existence theorem (see =-=[4]-=-): 6Theorem 4.1 (Grauert-Remmert) Let S be a connected normal complex space and T ⊂ S a proper closed analytical subset. Let Y := S −T, let X be a normal complex space and φ : X → Y be a ramified cov... |

12 |
Homotopieringe und Linsenräume
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(Show Context)
Citation Context ... to analytical ones). Hirzebruch [13] noticed that the abstract boundary of a bidimensional Hirzebruch-Jung singularity (Z, 0) of type An,q is a lens space L(n, q). As it was known since Reidemeister =-=[23]-=- that L(n, q) is homeomorphic to L(n ′ , q ′ ) if and only if n = n ′ and (q = q ′ or qq ′ ≡ 1(mod n)), this showed by proposition 7.4 that in this case the homeomorphism type of K(Z) determines the a... |

11 |
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(Show Context)
Citation Context ... C{X1, ..., Xd}[Y ] defining S such that its characteristic exponents A1, ..., AG verify: { (A 1 1 , ..., A 1 G ) ≥lex · · · ≥lex (A d 1 , ..., Ad G ) A 2 1 ̸= 0 or A 1 1 > 1 (12) Lipman [17] and Gau =-=[11]-=- showed that a sequence A1, ..., AG which verifies (12) - they called it then normalized - is an embedded topological invariant of (S, 0). In particular, it is an analytical invariant of (S, 0). In [2... |

11 |
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- 1965
(Show Context)
Citation Context ...er R(f) as a subset of ˜ C{X}. Moreover, we suppose that f is irreducible. Then all the differences of roots of f have dominating exponents, which are totally ordered for the componentwise order (see =-=[16]-=-, [17]). If G is their number, denote them by A1 < · · · < AG, Ai = (A1 i , ..., Adi ), ∀i ∈ {1, ..., G}. Definition 2.4 We call the vectors A1, ..., AG ∈ Q d + the characteristic exponents and the mo... |

10 |
Topological invariants of quasi-ordinary singularities
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(Show Context)
Citation Context ... (f) of f, which has therefore a dominating exponent. Definition 2.2 Let f ∈ C{X}[Y ] be unitary. If ∆Y (f) has a dominating exponent, we say that f is quasi-ordinary. The following theorem (see [1], =-=[17]-=-), generalizes the theorem of NewtonPuiseux for plane curves: Theorem 2.3 (Jung-Abhyankar) If f ∈ C{X}[Y ] is quasi-ordinary, then the set R(f) of roots of f embeds canonically in the algebra ˜ C{X}. ... |

8 |
Quasi-ordinary singularities via toric geometry, Tesis Doctoral, Universidad de La
- Pérez, D
- 2000
(Show Context)
Citation Context ...his let us introduce, following Lipman [17], the abelian groups M0 := Z d , Mi := Mi−1 + ZAi, ∀i ∈ {1, ..., G} and the successive indices Ni := (Mi : Mi−1), ∀i ∈ {1, ..., G}. Following González Pérez =-=[12]-=- we consider also the dual lattices Wk of the lattices Mk: Wk := Hom(Mk,Z), ∀ k ∈ {1, ..., G}. One has the inclusions: M0 � M1 � · · · � MG, W0 � W1 � · · · � WG. The following proposition was proved ... |

8 |
Reduction of the Singularities of an Algebraic Surface
- Walker
- 1935
(Show Context)
Citation Context ...anging the base of the initial projection using this desingularization morphism, he obtained a surface which is quasi-ordinary in the neighborhood of any of its points. This method was used by Walker =-=[25]-=- in order to prove the existence of a resolution of the singularities of a complex algebraic surface. This work is considered by Zariski [26] to be the first rigorous proof of this fact. Hirzebruch [1... |

7 |
Arbres de contact des singularités quasi-ordinaires et graphes d’adjacence pour les 3-variétés réelles, Thèse de doctorat
- Popescu-Pampu
- 2001
(Show Context)
Citation Context ...complex analytical space of arbitrary dimension is called a Hirzebruch-Jung singularity if it is analytically isomorphic with the normalization of an n-dimensional irreducible quasi-ordinary germ. In =-=[19]-=- (see also [21] and section 3) we showed that such a normalization is in fact analytically isomorphic to the germ at the 0-dimensional orbit of an affine toric variety defined by a maximal simplicial ... |

3 |
Singularities in the work of Friedrich Hirzebruch
- Brieskorn
- 2000
(Show Context)
Citation Context ...the numbers n, q starting from the self-intersection numbers of its components (see [3] and section 7). It is also known that this classification is topological. For historical details, see Brieskorn =-=[5]-=-. After the introduction of toric geometry in the ’70s, Hirzebruch-Jung surface singularities were seen to be precisely the germs analytically isomorphic to the germs of toric surfaces taken at 0-dime... |

3 | On the analytical invariance of the semigroups of a quasi-ordinary hypersurface singularity
- Popescu-Pampu
(Show Context)
Citation Context ...cal space of arbitrary dimension is called a Hirzebruch-Jung singularity if it is analytically isomorphic with the normalization of an n-dimensional irreducible quasi-ordinary germ. In [19] (see also =-=[21]-=- and section 3) we showed that such a normalization is in fact analytically isomorphic to the germ at the 0-dimensional orbit of an affine toric variety defined by a maximal simplicial pair. Conversel... |

3 |
Algebraic Surfaces Springer-Verlag 1934. AND FACTORIZATION OF BIRATIONAL MAPS 29 Department of Mathematics, Boston University, 111 Cummington
- Zariski
(Show Context)
Citation Context ...rks of Zariski and Lipman, the study of quasi-ordinary germs goes back at least to the work [14] of Jung on the problem of local uniformisation of surfaces. For details on it see the first chapter of =-=[26]-=-. The idea of Jung was to study an arbitrary germ of surface embedded in C 3 by considering a finite linear projection and an embedded desingularization of the discriminant curve. By changing the base... |

2 |
On Analytic Abelian Coverings
- Dimca
- 1988
(Show Context)
Citation Context ... extension of the universal covering map of the smooth part of (Z, 0) is called the orbifold map associated to (Z, 0). Denote by Γ(Z) the group of covering transformations of µ (in the terminology of =-=[8]-=-), formed by those analytical automorphisms φ : ( ˜ Z, 0) → ( ˜ Z, 0) which verify µ = µ ◦ φ. Consider its action: Γ(Z) ρ(Z) −→ GL(˜m/˜m 2 ) (1) on the Zariski cotangent space of ˜ Z at 0. Here ˜m den... |

1 | Normal quasi-ordinary singularities
- Aroca, Snoussi
- 2002
(Show Context)
Citation Context ..., σ0) = C d the canonical morphism associated to this change of lattice. We proved topologically the following theorem in [19] and [21]. A more algebraic proof was given later by Aroca and Snoussi in =-=[2]-=-. Theorem 3.1 One has the following commutative diagram, in which ν is a normalization morphism: ν (Z(W(ψ), σ0), 0) �� (S, 0) ������ � γ � W0 :W(ψ) ��� ψ ���������� (Cd, 0) In the special case in whic... |

1 |
Two-dimensional iterated torus knots and quasi-ordinary surface singularities
- Popescu-Pampu
(Show Context)
Citation Context ...ization algorithm 5.5 presented in the previous section. By using lemma 5.3, we can give in a more explicit form this algorithm, as we published it (but with slightly different notations) in [19] and =-=[20]-=-: Proposition 7.5 Let f ∈ C{X1, X2}[Y ] be an irreducible quasi-ordinary polynomial ( with characteristic exponents A1, ..., AG. If m(W0, σ0, ≺0; Wk) = k r1,1 rk 1,2 0 rk ) ( k s1,1 s , m(Wk−1, σk−1, ... |