On Auslander modules of normal surface singularities

Abstract We study thefundamental sequences of normal surface singularities. Our main result asserts that for rational singularities (with a technical side-condition) and for minimally elliptic singularities the middle termA, theAuslander module, is isomorphic to the module of Zariski differentials if and only if the singularity is quasihomogeneous.     

On normal surface singularities and a problem of enriques

We construct families of normal surface singularities with the following property: given any fiat projective connected family V →B of smooth, irreducible, minimal algebraic surfaces, the general singularity in one of our families cannot occur, analytically, on any algebraic surfaces which is Irrationally equivalent to a surface in V→B. In particular this holds for V→B consisting of a single rational surface, thus answering negatively to a long standing problem posed by F. Enriques. In order to prove the above mentioned results, wo develop a general, though elementary, method, based on the consideration of suitable correspondences, for comparing a given family of minimal surfaces with a family of surface singularities. Specifically the method in question gives us the possibility of comparing the parameters on which the two families depend, thus leading to the aforementioned results.     

On the singularities of weakly normal varieties

Among the weakly normal varieties (in the sense of Andreotti and Bombieri, [1]) are of particular interest those varieties such that the normalization morphism is unramified outside a subvariety of codimension not less than 2. We describe the singularities of these varieties (called here WN1) by means of analytic equations, tangent cones, analytic branches and we show that any irreducible projective variety is birationally equivalent to a WN1 hypersur face and that a Gorenstein variety is weakly normal if and only if it is WN1.This research was done when the authors were members of G.N.S.A.G.A. of the C.N.R.     

On normal two-dimensional double point singularities

Letq be a plane curve singularity and letp be the corresponding normal two-dimensional double point singularity. Let Γ and Γ₁ be the topological types of the minimal and of the canonical resolutions ofp respectively. An algorithm is given for finding the equisingular type ofq in terms of Γ₁. An algorithm is also given for finding all Γ₁ corresponding to a given Γ. There is at most one such Γ₁ in case Γ has no 1-cycles. This research was partially supported by the National Science Foundation. The author also is an Alfred P. Sloan Research Fellow.     

On equisingular deformations of cone-like surface singularities

In this note we study the rigidity-problem in the equisingular deformation theory for normal surface singularities whose exceptional sets of their minimal resolutions are smooth. We show that they admit non-trivial equisingular deformations if they are non-rational and if their analytic structures are not too different from those of cones. Latter condition is e.g. automatically satisfied if the absolute value of the selfintersection number of the exceptional set A is not less than the genus of A.     

On Cohen-Macaulay modules over non-commutative surface singularities

We generalize the results of Kahn about a correspondence between Cohen-Macaulay modules and vector bundles to non-commutative surface singularities. As an application, we give examples of non-commutative surface singularities which are not Cohen-Macaulay finite, but are Cohen-Macaulay tame.     

Reflexive modules on normal surface singularities and representations of the local fundamental group

We consider the Riemann-Hilbert correspondence on the complement of a normal surface singularity (X,x). Through a closure operation we obtain a correspondence between the category of finite dimensional representations of the local fundamental group and the category of left D_X,x-modules that are reflexive as O_X,x-modules. We show that under this correspondence profinite representations correspond to invariant modules and that these admit a canonical structure as left D_X,x-modules. We prove that the fundamental module is an invariant module if and only if (X,x) is a quotient singularity. Finally we investigate some algebraisation aspects.     

Removable singularities of CR functions on singular boundaries

The problem of analytic representation of integrable CR functions on hypersurfaces with singularities is treated. The nature of singularities does not matter while the set of singularities has surface measure zero. For simple singularities like cuspidal points, edges, corners, etc., also the behaviour of representing analytic functions near singular points is studied. Received: 8 December 2000; in final form: 24 June 2001/Published online: 1 February 2002     

Corner singularities between free surfaces and open boundaries

We consider the solution of the Stokes problem at a corner between a free surface and an inflow or outflow boundary. A formal asymptotic solution for the dominant contribution to the streamfunction near the corner is derived. We give a heuristic discussion of the relevance of the nature of the corner singularity to the formulation of well-posed boundary value problems.     

Minimizing singularities of generic plane disks with immersed boundaries

A cooriented circle immersion into the plane can be extended to a stable map of the disk which is an immersion in a neighborhood of the boundary and with outward normal vector field along the boundary equal to the given coorienting normal vector field. We express the minimal number of fold components of such a stable map as a function of its number of cusps and of the normal degree of its boundary. We also show that this minimum is attained for any cooriented circle immersion of normal degree not equal to one. The first author is a research fellow of the Royal Swedish Academy of Sciences sponsored by the Knut and Alice Wallenberg foundation.     

Quasi-determinantal rational surface singularities

In this paper we give explicit equations for quasi-determinantal rational surface singularities, extending previous results for determinantal rational surface singularities.