Dmitry Kerner

Enumeration of uni-singular algebraic hypersurfaces

We enumerate complex algebraic hypersurfaces in

On the Enumeration of Complex Plane Curves with Two Singular Points

P^n

Determinantal variety and normal embedding

, 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 equi-singular strata in the parameter space of hypersurfaces. We suggest an inductive procedure, based on 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 details the generalized Newton-non-degenerate singularities. We give also examples of enumeration in some other cases.

A strong version of Implicit Function Theorem

We study equisingular strata of plane curves with two singular points of prescribed types. The method of the previous work [25] is generalized to this case. In particular, we consider the enumerative problem for plane curves with two singular points of linear singularity types. First, the problem for two ordinary multiple points of fixed multiplicities is solved. Then the enumeration for arbitrary linear types is reduced to the case of ordinary multiple points and to the understanding of “merging” of singular points. Many examples and numerical answers are given.

Durfee-type bound for some non-degenerate complete intersection singularities

The space of matrices of positive determinant GL^+_n inherits an extrinsic metric space structure from R^{n^2}. On the other hand, taking the infimum of the lengths of all paths connecting two points in GL^+_n gives an intrinsic metric. We prove bilipschitz equivalence for intrinsic and extrinsic metrics on GL^+_n, exploiting the conical structure of the stratification of the space of n by n matrices by rank.

A generalized FKG-inequality for compositions☆

We suggest the necessary/sufficient criteria for existence of a (order-by-order) solution

Finite determinacy of matrices over local rings.II. Tangent modules to the miniversal deformations for group-actions involving the ring automorphisms

{y}({x})

Decomposability of local determinantal representations of hypersurfaces

y(x) of a functional equation