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**1 - 6**of**6**### Enumeration of uni-singular 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 Newton-non-degenerate singularities. We also give examples of enumeration in some other cases.

### On the geometry of some strata of uni-singular curves

, 2008

"... We study geometric properties of linear strata of uni-singular 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 uni-singular 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

### 1 On the enumeration of complex plane curves with two singular points

, 2008

"... We study equi-singular 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 equi-singular 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. MSC: primary-14N10, 14N35 secondary-14H10, 14H50

### 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 d-dimensional 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 d-dimensional 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 d-dimensional manifolds with prescribed singularities, and explain the relation of singular cobordism categories to the bordism version of the Gromov h-principle. 1.

### 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 n-dimensional 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 n-dimensional 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 n-dimensional 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 n-dimensional 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.