Results 1  10
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11
Thom series of contact singularities
, 2008
"... Abstract. Thom polynomials measure how global topology forces singularities. The power of Thom polynomials predestine them to be a useful tool not only in differential topology, but also in algebraic geometry (enumerative geometry, moduli spaces) and algebraic combinatorics. The main obstacle of the ..."
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Abstract. Thom polynomials measure how global topology forces singularities. The power of Thom polynomials predestine them to be a useful tool not only in differential topology, but also in algebraic geometry (enumerative geometry, moduli spaces) and algebraic combinatorics. The main obstacle of their widespread application is that only a few, sporadic Thom polynomials have been known explicitly. In this paper we develop a general method for calculating Thom polynomials of singularities. Along the way, relations with the equivariant geometry of (punctual, local) Hilbert schemes, and with iterated residue identities are revealed. 1.
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 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
COBORDISMS OF MAPS WITH SINGULARITIES OF A GIVEN CLASS
, 2007
"... Let P be a smooth manifold of dimension p. We will describe the group of all cobordism classes of smooth maps of ndimensional closed manifolds into P with singularities of a given class in terms of certain stable homotopy groups by applying the homotopy principle on the existence level, which is a ..."
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Let P be a smooth manifold of dimension p. We will describe the group of all cobordism classes of smooth maps of ndimensional closed manifolds into P with singularities of a given class in terms of certain stable homotopy groups by applying the homotopy principle on the existence level, which is assumed to hold for those smooth maps. We will also deal with the oriented version and construct a classifying space of this oriented cobordism group in the dimensions n < p.
COBORDISMS OF MAPS WITH SINGULARITIES OF A GIVEN CLASS
, 2008
"... Let P be a smooth manifold of dimension p. We will describe the group of all cobordism classes of smooth maps of ndimensional closed manifolds into P with singularities of a given class (including all fold singularities if n ≧ p) in terms of certain stable homotopy groups by applying the homotopy p ..."
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Let P be a smooth manifold of dimension p. We will describe the group of all cobordism classes of smooth maps of ndimensional closed manifolds into P with singularities of a given class (including all fold singularities if n ≧ p) in terms of certain stable homotopy groups by applying the homotopy principle on the existence level, which is assumed to hold for those smooth maps. We will also deal with the oriented version and construct a classifying space of this oriented cobordism group in the dimensions n < p and n ≧ p ≧ 2.
THE GENERAL QUADRUPLE POINT FORMULA
"... Abstract. Maps between manifolds M m → N m+ℓ (ℓ>0) have multiple points, and more generally, multisingularities. The closure of the set of points where the map has a particular multisingularity is called the multisingularity locus. There are universal relations among the cohomology classes repres ..."
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Abstract. Maps between manifolds M m → N m+ℓ (ℓ>0) have multiple points, and more generally, multisingularities. The closure of the set of points where the map has a particular multisingularity is called the multisingularity locus. There are universal relations among the cohomology classes represented by multisingularity loci, and the characteristic classes of the manifolds. These relations include the celebrated Thom polynomials of monosingularities. For multisingularities, however, only the form of these relations is clear in general (due to Kazarian), the concrete polynomials occurring in the relations are much less known. In the present paper we prove the first general such relation outside the region of Morinmaps: the general quadruple point formula. We apply this formula in enumerative geometry by computing the number of 4secant linear spaces to smooth projective varieties. Some other multisingularity formulas are also studied, namely 5, 6, 7 tuple point formulas, and one corresponding to Σ 2 Σ 0 multisingularities. 1. Introduction. Let f: M m → N n be a holomorphic map between compact complex manifolds, and let ℓ = n − m> 0. Associated with a list of singularities α =(α1,..., αr) one can consider the following multisingularity locus in the source manifold M: the collection of points x where the map has singularity α1,
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.
Schubert Calculus according to Schubert
, 2008
"... We try to understand and justify Schubert calculus the way Schubert did it. This is the english, extended version of a previously posted preprint math.AG/0409281. Contents ..."
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We try to understand and justify Schubert calculus the way Schubert did it. This is the english, extended version of a previously posted preprint math.AG/0409281. Contents
SINGULAR COBORDISM CATEGORIES
, 804
"... Abstract. Recently Galatius, Madsen, Tillmann and Weiss identified the homotopy type of the classifying space of the cobordism category of embedded ddimensional manifolds [7] for each positive integer d. Their result lead to a new proof of the generalized standard Mumford conjecture. We extend the ..."
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Abstract. Recently Galatius, Madsen, Tillmann and Weiss identified the homotopy type of the classifying space of the cobordism category of embedded ddimensional manifolds [7] for each positive integer d. Their result lead to a new proof of the generalized standard Mumford conjecture. We extend the main theorem of [7] to the case of cobordism categories of embedded ddimensional manifolds with prescribed singularities, and explain the relation of singular cobordism categories to the bordism version of the Gromov hprinciple. 1.