Equisingular Deformations Research Articles

Obstructions to deforming maps from curves to surfaces

This paper studies the obstructions to deforming a map from a complex variety to another variety which is an immersion of codimension one. We extend the classical notion of semiregularity of subvarieties to maps between varieties, and prove the unobstructedness of the deformation of such a map, even when the image is non-reduced. As an application, we give a simple but effective criterion for the vanishing of the obstructions to equisingular deformations of nodal curves on surfaces.

Deformations of plane curves and Jacobian syzygies

AbstractWe relate the equianalytic and the equisingular deformations of a reduced complex plane curve to the Jacobian syzygies of its defining equation. Several examples and conjectures involving rational cuspidal curves are discussed.

Topologically equisingular deformations of homogeneous hypersurfaces with line singularities are equimultiple

We prove that if [Formula: see text] is a family of line singularities with constant Lê numbers and such that [Formula: see text] is a homogeneous polynomial, then [Formula: see text] is equimultiple. This extends to line singularities a well-known theorem of Gabrièlov and Kušnirenko concerning isolated singularities. As an application, we show that if [Formula: see text] is a topologically [Formula: see text]-equisingular family of line singularities, with [Formula: see text] homogeneous, then [Formula: see text] is equimultiple. This provides a new partial positive answer to the famous Zariski multiplicity conjecture for a special class of non-isolated hypersurface singularities.

Classification of simple quartics up to equisingular deformation

We study complex spatial quartic surfaces with simple singularities up to equisingular deformations; as a first step, give a complete equisingular deformation classification of non-special simple quartic surfaces.

Logarithmic residues along plane curves

Let (D,0)⊂(C2,0) be a plane curve germ defined by a reduced equation f. We prove that a fractional ideal I of D satisfies a symmetry property with its dual, and then apply it to study the behavior of the module of logarithmic residues of D in equisingular deformations.

SOME SPLICE QUOTIENT DOUBLE POINTS

The splice quotients are an interesting class of normal surface singularities with rational homology sphere links. In general, it can be difficult to determine whether or not a singularity is a splice quotient (an analytic condition). We consider splice quotient deformations of splice quotients of the form z2= xa+ yb, and show that in general not all equisingular deformations are splice quotients.

On deformations of singular plane sextics

We study complex plane projective sextic curves with simple singularities up to equisingular deformations. It is shown that two such curves are deformation equivalent if and only if the corresponding pairs are diffeomorphic. A way to enumerate all deformation classes is outlined, and a few examples are considered, including classical Zariski pairs; in particular, promising candidates for homeomorphic but not diffeomorphic pairs are found.

Equisingular deformations of plane curves in arbitrary characteristic

The purpose of this paper is to develop the theory of equisingular deformations of plane curve singularities in arbitrary characteristic. We study equisingular deformations of the parametrization and of the equation and show that the base space of its semiuniversal deformation is smooth in both cases. Our approach through deformations of the parametrization is elementary and we show that equisingular deformations of the parametrization form a linear subfunctor of all deformations of the parametrization. This gives additional information, even in characteristic zero, the case which was treated by J. Wahl. The methods and proofs extend easily to good characteristic, that is, when the characteristic does not divide the multiplicity of any branch of the singularity. In bad characteristic, however, new phenomena occur and we are naturally led to consider weakly trivial (respectively, weakly equisingular) deformations, that is, deformations which become trivial (respectively, equisingular) after a finite and dominant base change. The semiuniversal base space for weakly equisingular deformations is, in general, not smooth but becomes smooth after a finite and purely inseparable base extension. The proof of this fact is more complicated and we introduce new constructions which may have further applications in the theory of singularities in positive characteristic.

Equisingular Deformations of Sandwiched Singularities

We relate the equisingular deformation theory of plane curve singularities and sandwiched surface singularities. We show the existence of a smooth map between the two corresponding deformation functors and study the kernel of this map. In particular we show that the map is an isomorphism when a certain invariant is large enough.

On the freeness of equisingular deformations of plane curve singularities

We consider surface singularities in C 3 arising as the total space of an equisingular deformation of an isolated curve singularity of the form f( x, y)+ zg( x, y) with f and g weighted homogeneous. We give a criterion that such a surface is a free divisor in the sense of Saito. We deduce that the Hessian deformation defines a free divisor for nonsimple weighted homogeneous singularities, and that the failure of this property “almost” characterizes the simple singularities. The criterion also yields distinct deformations of the same curve singularity, exactly one of which is free, showing that freeness is not a topological property.

On the freeness of equisingular deformations of plane curve singularities

On the freeness of equisingular deformations of plane curve singularities

Taut, pseudotaut and equisingularly rigid singularities

The aim of this paper is to find all plane curve singularities that are taut resp. pseudotaut. It turns out that this problem coincides with the determination of equisingularly rigid singularities. The latter one is achieved in the irreducible case by explicit construction of nontrivial deformations usiing analytical invariants of the Puiseux expansion introduced by Kasner and Zariski, in the reducible case with a cohomological criterion for the triviality of Wahl's functor ES of equisingular deformations of a resolution. Equisingular rigidity is the same as K-zero- or unimodality with discrete parameter. An application is the determination of all equisingularly rigid double points of surfaces, which are just the stabilizations of equisingularly rigid plane curve singularities.

Equisingular deformations of isolated 2-dimensional hypersurface singularities

Equisingular deformations of isolated 2-dimensional hypersurface singularities

An Equisingular Deformation Theory via Embedded Resolution of Singularities

In this paper we shall give a definition of equisingular deformations of isolated singularities and prove some results about it such as the existence of semi-universal deformations. Roughly speaking, equisingularity means in this paper the existence of a simultaneous embedded resolution. Of course, in the case of plane curve singularities the definition coincides with Zariski's classical one. In Section 1 we shall define equisingular deformation and prove some elementary properties of it. In Section 2 we shall study the deformation of locally product type of the pair obtained by the embedded resolution of singularity. In Section 3 we shall prove the existence theorem of semi-universal deformations. The author would like to express his heartfelt thanks to Professor S. litaka, the conversations with whom were very fruitful and encouraging, and to Pf. M. Merle who pointed out to the author Example 2.

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 equisingular deformations of plane curve singularities

On equisingular deformations of plane curve singularities

Equisingular Deformations of Normal Surface Singularities, I

Equisingular Deformations of Normal Surface Singularities, I

Equisingular deformations of Puiseux expansions

Parametrizations of a deformation (over a complete local ring) of an irreducible algebroid curve are studied. With these parametrizations another definition of equisingularity (equivalent to the known ones) is given, by using their characteristic numbers. These methods are also used in the complex analytic case. The following applications of these techniques are given: a proof of the existence of a versal equisingular deformation in the complex analytic case; a proof that an equisingular formal family of branches is determined by its ν \nu -truncation ( ν \nu large enough, depending only on the characteristic of the special fiber).

Equisingular Deformations of Plane Algebroid Curves

We construct a formal versal equisingular deformation of a plane algebroid curve (in characteristic zero), and show it is smoothly embedded in the whole deformation space of the singularity. Closer analysis relates equisingular deformations of the curve to locally trivial deformations of a certain (nonreduced) projective curve. Finally, we prove that algebraic ${\pi _1}$ of the complement of a plane algebroid curve remains constant during formal equisingular deformation.

Equisingular deformations of plane algebroid curves

We construct a formal versal equisingular deformation of a plane algebroid curve (in characteristic zero), and show it is smoothly embedded in the whole deformation space of the singularity. Closer analysis relates equisingular deformations of the curve to locally trivial deformations of a certain (nonreduced) projective curve. Finally, we prove that algebraic π 1 {\pi _1} of the complement of a plane algebroid curve remains constant during formal equisingular deformation.