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11
Positivity of Schur function expansions of Thom polynomials
"... (10.05.2006; revised 17.05.2006) Combining the Kazarian approach to Thom polynomials via classifying spaces of singularities with the FultonLazarsfeld theory of numerical positivity for ample vector bundles, we show that the coefficients of Schur function expansions of the Thom polynomials of stabl ..."
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Cited by 7 (5 self)
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(10.05.2006; revised 17.05.2006) Combining the Kazarian approach to Thom polynomials via classifying spaces of singularities with the FultonLazarsfeld theory of numerical positivity for ample vector bundles, we show that the coefficients of Schur function expansions of the Thom polynomials of stable singularities are nonnegative. 1
Thom polynomials and Schur functions: towards the singularities Ai(−)
, 2008
"... We develop algebrocombinatorial tools for computing the Thom polynomials for the Morin singularities Ai(−) (i ≥ 0). The main tool is the function F (i) r defined as a combination of Schur functions with certain numerical specializations of Schur polynomials as their coefficients. We show that the T ..."
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Cited by 4 (3 self)
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We develop algebrocombinatorial tools for computing the Thom polynomials for the Morin singularities Ai(−) (i ≥ 0). The main tool is the function F (i) r defined as a combination of Schur functions with certain numerical specializations of Schur polynomials as their coefficients. We show that the Thom polynomial T Ai for the singularity Ai (any i) associated with maps (C •,0) → (C •+k,0), with any parameter k ≥ 0, under the assumption that Σ j = ∅ for all j ≥ 2, is given by F (i) k+1. Equivalently, this says that “the 1part ” of T Ai (i) equals F k+1. We investigate 2 examples when T Ai apart from its 1part consists also of the 2part being a single Schur function with some multiplicity. Our computations combine the characterization of Thom polynomials via the “method of restriction equations” of Rimányi et al. with the techniques of Schur functions.
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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Cited by 1 (1 self)
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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 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 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.
Enumeration of unisingular algebraic hypersurfaces
, 2007
"... We enumerate complex algebraic hypersurfaces in P n, of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equisingular strata in the parameter space of hypersurfaces. We suggest an inductive procedure ..."
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We enumerate complex algebraic hypersurfaces in P n, of a given (high) degree with one singular point of a given singularity type. Our approach is to compute the (co)homology classes of the corresponding equisingular strata in the parameter space of hypersurfaces. We suggest an inductive procedure, based on an intersection theory combined with liftings and degenerations. The procedure computes the (co)homology class in question, whenever a given singularity type is properly defined and the stratum possesses good geometric properties. We consider in detail the generalized Newtonnondegenerate singularities. We also give examples of enumeration in some other cases.
FOLD MAPS, FRAMED IMMERSIONS AND SMOOTH STRUCTURES
, 803
"... Abstract. We show that the cobordism group of fold maps of even nonpositive codimension q into a manifold N is a sum of q/2 cobordism groups of framed immersions into N and a group related to diffeomorphism groups of manifolds of dimension q + 1. In the case of maps of odd nonpositive codimension ..."
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Abstract. We show that the cobordism group of fold maps of even nonpositive codimension q into a manifold N is a sum of q/2 cobordism groups of framed immersions into N and a group related to diffeomorphism groups of manifolds of dimension q + 1. In the case of maps of odd nonpositive codimension q, we show that the cobordism groups of fold maps split off (q − 1)/2 cobordism groups of framed immersions. 1.