Patrick Popescu-Pampu

On the analytical invariance of the semigroups of a quasi-ordinary hypersurface singularity

We associate to any irreducible germ S of complex quasi-ordinary hypersurface an analytically invariant semigroup. We deduce a direct proof (without passing through their embedded topological invariance) of the analytical invariance of the normalized characteristic exponents. These exponents generalize the generic Newton-Puiseux exponents of plane curves. Incidentally, we give a toric description of the normalization morphism of the germ S.

On the Milnor fibres of cyclic quotient singularities

The oriented link of the cyclic quotient singularity Xp,q is orientation-preserving diffeomorphic to the lens space L(p, q) and carries the standard contact structure ξst. Lisca classified the Stein fillings of (L(p, q) ,ξ st) up to diffeomorphisms and conjectured that they correspond bijectively through an explicit map to the Milnor fibres associated with the irreducible components (all of them being smoothing components) of the reduced miniversal space of deformations of Xp,q. We prove this conjecture using the smoothing equations given by Christophersen and Stevens. Moreover, based on a different description of the Milnor fibres given by de Jong and van Straten, we also canonically identify these fibres with Lisca’s fillings. Using these and a newly introduced additional structure (the order) associated with lens spaces, we prove that the above Milnor fibres are pairwise non-diffeomorphic (by diffeomorphisms which preserve the orientation and order). This also implies that de Jong and van Straten parametrize in the same way the components of the reduced miniversal space of deformations as Christophersen and Stevens.

Numerically Gorenstein surface singularities are homeomorphic to Gorenstein ones

Consider a normal complex analytic surface singularity. It is called Gorenstein if the canonical line bundle is holomorphically trivial in some punctured neighborhood of the singular point and is called numerically Gorenstein if this line bundle is topologically trivial. The second notion depends only on the topological type of the singularity. Laufer proved in 1977 that, given a numerically Gorenstein topological type of singularity, every analytical realization of it is Gorenstein if and only if one has either a Kleinian or a minimally elliptic topological type. The question to know if any numerically Gorenstein topology was realizable by some Gorenstein singularity was left open. We prove that this is indeed the case. Our method is to plumb holomorphically meromorphic 2-forms obtained by adequate pull-backs of the natural holomorphic symplectic forms on the total spaces of the canonical line bundles of complex curves. More generally, we show that any normal surface singularity is homeomorphic to a Q-Gorenstein singularity whose index is equal to the smallest common denominator of the coefficients of the canonical cycle of the starting singularity.

Two-dimensional iterated torus knots and quasi-ordinary surface singularities

Abstract We define a notion of 2-dimensional iterated torus knot, namely special embeddings of a 2-torus in the Cartesian product of a 2-torus and a 2-disc. We apply this definition to give a description of the embedded topology of the boundary of an irreducible quasi-ordinary hypersurface germ of dimension 2, in terms of the characteristic exponents of an arbitrary quasi-ordinary projection. Incidentally, we give an algorithm for computing the Jung–Hirzebruch type of its normalization. To cite this article: P. Popescu-Pampu, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

Le cerf-volant d'une constellation

Consider a smooth point O of a complex analytic surface S. A constellation based at O is a set of infinitely near points of O, centers of a sequence of blow-ups above O. Finite constellations are usually encoded in two ways: either using an Enriques diagram, or using the dual graph of the divisor obtained by blowing-up the points of the constellation. Both are decorated trees which encode completely the combinatorics of the constellation. Algorithms of passage from one to the other are known, but they do not allow to get a geometrical picture of their relation. We associate to a constellation a geometrical simplicial complex of dimension two, called its kite, endowed with an affine structure, and we prove that it contains canonically both the Enriques diagram and the dual graph. Moreover, the decorations of the two trees may be read very easily on the affine geometry of the kite. This allows to understand geometrically the relations between the graphs, as well as their relation with the valuative tree of Favre and Jonsson, which may be interpreted as the dual graph of the constellation of all the points infinitely near O. In fact, the kites of finite constellations get glued into an infinite kite endowed with a 1-dimensional foliation, whose space of leaves is the valuative tree. The transition towards the computations with continued fractions is ensured by partial embeddings of the kites into a simplicial complex canonically associated to a base of a lattice, called its lotus. This last notion is briefly explored in any dimension.

SUR LE CONTACT D’UNE HYPERSURFACE QUASI-ORDINAIRE AVEC SES HYPERSURFACES POLAIRES

Resume Nous appelons polynome quasi-ordinaire de Laurent un polynome unitaire f(Y ) dont les coefficients sont des series de Laurent a plusieurs variables et tel que son discriminant soit le produit d’un monome de Laurent et d’une serie entiere de terme constant non-nul. Si la derivee ∂f/∂Y rendue unitaire est encore quasi-ordinaire de Laurent—ce qui peut etre toujours obtenu par changement de base—nous montrons que l’on peut mesurer le contact de ses facteurs avec ceux de f en fonction d’invariants discrets de f qui mesurent le contact entre ses racines, codes sous la forme de l’arbre d’Eggers–Wall. Tous les calculs sont faits en termes de châines et de cochâines supportees par cet arbre. Ce travail constitue une generalisation de resultats connus pour les germes de courbes planes.

Ultrametric Spaces of Branches on Arborescent Singularities

Let S be a normal complex analytic surface singularity. We say that S is arborescent if the dual graph of any good resolution of it is a tree. Whenever A, B are distinct branches on S, we denote by A ⋅ B their intersection number in the sense of Mumford. If L is a fixed branch, we define UL(A, B) = (L ⋅ A)(L ⋅ B)(A ⋅ B)−1 when A ≠ B and UL(A, A) = 0 otherwise. We generalize a theorem of Ploski concerning smooth germs of surfaces, by proving that whenever S is arborescent, then UL is an ultrametric on the set of branches of S different from L. We compute the maximum of UL, which gives an analog of a theorem of Teissier. We show that UL encodes topological information about the structure of the embedded resolutions of any finite set of branches. This generalizes a theorem of Favre and Jonsson concerning the case when both S and L are smooth. We generalize also from smooth germs to arbitrary arborescent ones their valuative interpretation of the dual trees of the resolutions of S. Our proofs are based in an essential way on a determinantal identity of Eisenbud and Neumann.

On the Milnor Fibers of Sandwiched Singularities

The sandwiched surface singularities are those rational surface singulari- ties which dominate birationally smooth surface singularities. de Jong and van Straten showed that one can reduce the study of the deformations of a sandwiched surface sin- gularity to the study of deformations of a 1-dimensional object, a so-called decorated plane curve singularity. In particular, the Milnor fibers corresponding to their various smoothing components may be reconstructed up to diffeomorphisms from those defor- mations of associated decorated curves which have only ordinary singularities. Part of the topology of such a deformation is encoded in the incidence matrix between the irre- ducible components of the deformed curve and the points which decorate it, well-defined up to permutations of columns. Extending a previous theorem ofours, which treated the case of cyclic quotient singularities, we show that the Milnor fibers which correspond to deformations whose incidence matrices are different up to permutations of columns are not diffeomorphic in a strong sense. This gives a lower bound on the number of Stein fillings of the contact boundary of a sandwiched singularity.

On the generalized Nash problem for smooth germs and adjacencies of curve singularities

Abstract In this paper we explore the generalized Nash problem for arcs on a germ of smooth surface: given two prime divisors above its special point, to determine whether the arc space of one of them is included in the arc space of the other one. We prove that this problem is combinatorial and we explore its relation with several notions of adjacency of plane curve singularities.

Generalized Plumbings and Murasugi Sums

We propose a generalization of the classical notions of plumbing and Murasugi summing operations to smooth manifolds of arbitrary dimensions, so that in this general context Gabai’s credo “the Murasugi sum is a natural geometric operation” holds. In particular, we prove that the sum of the pages of two open books is again a page of an open book and that there is an associated summing operation of Morse maps. We conclude with several open questions relating this work with singularity theory and contact topology.