arXiv subjects

Dmitry Kerner

Publications and source records attributed to Dmitry Kerner.

45 records · Page 3Linked to original sources

arXiv2010

On the enumeration of complex plane curves with two singular points

We study equi-singular strata of plane curves with two singular points of prescribed types. The method of the previous work [Kerner06] 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.

math.AG↗

arXiv2010

On some generalizations of Newton non degeneracy for hypersurface singularities

We introduce two generalizations of Newton-non-degenerate (Nnd) singularities of hypersurfaces. Roughly speaking, an isolated hypersurface singularity is called topologically Newton-non-degenerate (tNnd) if the local embedded topological singularity type can be restored from a collection of Newton diagrams (for some coordinate choices). A singularity that is not tNnd is called essentially Newton-degenerate. For plane curves we give an explicit characterization of tNnd singularities; for hypersurfaces we provide several examples. Next, we treat the question: whether Nnd or tNnd is a property of singularity types or of particular representatives. Namely, is the non-degeneracy preserved in an equi-singular family? This fact is proved for curves. For hypersurfaces we give an example of a Nnd hypersurface whose equi-singular deformation consists of essentially Newton-degenerate hypersurfaces. Finally, we define the directionally Newton-non-degenerate germs, a subclass of tNnd germs. For such singularities the classical formulas for the Milnor number and the zeta function of the Nnd hypersurface are generalized.

math.AG↗

arXiv2009

On the collisions of singular points of complex algebraic plane curves

We study the "generic" degenerations of curves with two singular points when the points merge. First, the notion of generic degeneration is defined precisely. Then a method to classify the possible results of generic degenerations is proposed in the case of linear singularity types. We discuss possible bounds on the singularity invariants of the resulting type in terms of the initial types. In particular the strict upper bound on the resulting multiplicity is proved and a sufficient condition for $δ=const$ collision is given.

math.AG↗

arXiv2009

On the equi-normalizable deformations of singularities of complex plane curves

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 equi-normalizable or equi-generic 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 equi-normalizable families. We consider in details deformations of ordinary multiple points, 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 $A_k$'s.

math.AG↗

arXiv2009

Enumeration of uni-singular algebraic hypersurfaces

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 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.

math.AG↗

arXiv2008

Automorphisms of $P_8$ singularities and the complex crystallographic groups

The paper completes the study of symmetries of parabolic function singularities with relation to complex crystallographic groups that was started in \cite{GM,X9}. We classify smoothable automorphisms of $P_8$ singularities which split the kernel of the intersection form on the second homology. For such automorphisms, the monodromy groups acting on the duals to the eigenspaces with degenerate intersection form are then identified as some of complex affine reflection groups tabled in \cite{P}.

math.AG↗

arXiv2007

On the geometry of some strata of uni-singular curves

We study geometric properties of linear strata of uni-singular curves. The singularities of closures of the strata are resolved and the resolutions are represent as projective bundles. This enables 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 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 curves with two singular points, one of them being a node.

math.AG↗

arXiv2006

Enumeration of singular algebraic curves

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 strata in the parameter spaces of plane curves. We suggest a recursive procedure, which is based on the intersection theory combined with liftings and degenerations. The procedure computes the homology class in question whenever a given singularity type is defined. Our method does not require the knowledge of all the possible deformations of a given singularity.

math.AG↗

arXiv2001

Pseudoconformal field theory at the "wrong level"

Pseudoaffine theories are characterized by formal replacement of the level to the fractional number: $k\to\frac{k}{q}$, where $q$ is integer strange to $k(g+k)$ ($g$ - dual Coxeter number). An example of "forbidden" $q$ is considered (SU(2), $q=2$). The fusions of obtained theory are similar to the affine ones. Spectra of minimal models are calculated.

hep-th↗