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Enumeration of singular algebraic curves
, 2005
"... We enumerate plane complex algebraic curves of a given degree with one singularity of any given topological type. Our approach is to compute the homology classes of the corresponding equisingular families in the parameter spaces of plane curves. We suggest an inductive procedure, which is based on t ..."
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We enumerate plane complex algebraic curves of a given degree with one singularity of any given topological type. Our approach is to compute the homology classes of the corresponding equisingular families in the parameter spaces of plane curves. We suggest an inductive procedure, which is based on the intersection theory combined with liftings and degenerations, and which computes the homology class in question whenever a given singularity type is defined. Our method does not require the knowledge of all possible deformations of a given singularity as it was in
On the enumeration of complex plane curves with two singular points
, 2008
"... We study equisingular strata of curves with two singular points of prescribed types. The method of the previous work [Kerner04] is generalized to this case. This allows to solve the enumerative problem for plane curves with two singular points of linear singularity types. In the general case this r ..."
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We study equisingular strata of curves with two singular points of prescribed types. The method of the previous work [Kerner04] is generalized to this case. This allows to solve the enumerative problem for plane curves with two singular points of linear singularity types. In the general case this reduces the enumerative questions to the problem of collision of the two singular points. The method is applied to several cases, e.g. enumeration of curves with two ordinary multiple points, with a point of a linear singularity type and a node etc. Explicit numerical results are given. An elementary application of the method is the determination of Thom polynomials for curves with one singular point (for some series of singularity types). Some examples are given.
On
, 708
"... the collisions of singular points of complex algebraic plane curves ..."
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On
, 708
"... the collisions of singular points of complex algebraic plane curves ..."
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On the geometry of some strata of unisingular curves
, 2008
"... We study geometric properties of linear strata of unisingular curves. We resolve the singularities of closures of the strata and represent the resolutions as projective bundles. This enables us to study their geometry. In particular we calculate the Picard groups of the strata and the intersection ..."
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We study geometric properties of linear strata of unisingular curves. We resolve the singularities of closures of the strata and represent the resolutions as projective bundles. This enables us to study their geometry. In particular we calculate the Picard groups of the strata and the intersection rings of the closures of the strata. The rational equivalence classes of some geometric cycles on the strata are calculated. As an application we give an example when the proper stratum is not affine. As an auxiliary problem we discuss the collision of two singular points, restrictions on possible resulting singularity types and solve the collision problem in several cases. Then we present some cases of enumeration of
On the equinormalizable deformations of singularities of complex plane curves.
, 805
"... Abstract. We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total δ invariant is preserved. These are also known as equinormalizable or equigeneric deformations. We restrict primarily to the deformations of sing ..."
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Abstract. We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total δ invariant is preserved. These are also known as equinormalizable or equigeneric deformations. We restrict primarily to the deformations of singularities with smooth branches. A natural invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in equinormalizable families. We consider in details deformations of ordinary multiple point, the deformations of a singularity into the collections of ordinary multiple points and deformations of the type x p + y pk into the collections of Ak’s. Contents
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, 805
"... On the δ = const, κ = const deformations/degenerations of singularities of complex plane curves. ..."
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On the δ = const, κ = const deformations/degenerations of singularities of complex plane curves.
On the δ = const deformations/degenerations of singularities of complex plane curves.
, 805
"... Abstract. We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total δ invariant is preserved (aka equigeneric deformations). We restrict primarily to the deformations of singularities with smooth branches. A new in ..."
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Abstract. We study a specific class of deformations of curve singularities: the case when the singular point splits to several ones, such that the total δ invariant is preserved (aka equigeneric deformations). We restrict primarily to the deformations of singularities with smooth branches. A new invariant of the singular type is introduced: the dual graph. It imposes severe restrictions on the possible collisions/deformations. And allows to prove some bounds on the variation of classical invariants in collisions. We consider in details the δ = const deformations of ordinary multiple point, the deformation of a singularity into the collection of ordinary multiple points and the deformation of the type x p + y pk into a collection of Ak’s. 1.