Alexander modules of irreducible CC-groups

A complete description of the Alexander modules of knotted $n$-manifolds in the sphere $S^{n+2}$, $n\geq 2$, and irreducible Hurwitz curves is given. This description is applied to investigate properties of the first homology groups of cyclic coverings of the sphere $S^{n+2}$ and the projective complex plane $\mathbb C\mathbb P^2$ branched respectively alone knotted $n$-manifolds and along irreducible Hurwitz (in particular, algebraic) curves.Comment: 44 page

A remark on the non-rationality Problem for generic cubic fourfolds

It is proved that the non-rationality of a generic cubic fourfold follows from a conjecture on the non-decomposability in the direct sum of non-trivial polarized Hodge structures of the polarized Hodge structure on transcendental cycles on a projective surface.Comment: 9 page

On the almost generic covers of the projective plane

A finite morphism $f:X\to \mathbb P^2$ of a a smooth irreducible projective surface $X$ is called an almost generic cover if for each point $p\in \mathbb P^2$ the fibre $f^{-1}(p)$ is supported at least on $deg(f)-2$ distinct points and $f$ is ramified with multiplicity two at a generic point of its ramification locus $R$. In the article, the singular points of the branch curve $B\subset\mathbb P^2$ of an almost generic cover are investigated and main invariants of the covering surface $X$ are calculated in terms of invariants of the curve $B$.Comment: 13 pages. Submitted to the Pure and Applied Math. Quarterl

Jacobian Conjecture and Nilpotent Mappings

We prove the equivalence of the Jacobian Conjecture (JC(n)) and the Conjecture on the cardinality of the set of fixed points of a polynomial nilpotent mapping (JN(n)) and prove a series of assertions confirming JN(n).Comment: 10 pages, LaTeX2

On a Chisini Conjecture

Chisini's conjecture asserts that for a cuspidal curve $B\subset \mathbb P^2$ a generic morphism $f$ of a smooth projective surface onto $\mathbb P^2$ of degree $\geq 5$, branched along $B$, is unique up to isomorphism. We prove that if $\deg f$ is greater than the value of some function depending on the degree, genus, and number of cusps of $B$, then the Chisini conjecture holds for $B$. This inequality holds for many different generic morphisms. In particular, it holds for a generic morphism given by a linear subsystem of the $m$th canonical class for almost all surfaces with ample canonical class.Comment: 28 pages, LaTeX2

On Chisini's Conjecture. II

It is proved that if $S\subset \mathbb P^N$ is a smooth projective surface and $f:S\to \mathbb P^2$ is a generic linear projection branched over a cuspidal curve $B\subset \mathbb P^2$, then the surface $S$ is determined uniquely up to an isomorphism of $S$ by the curve $B$.Comment: 14 page

On the monodromy of the inflection points of plane curves

We prove that the monodromy group of the inflection points of plane curves of degree $d$ is the symmetric group $\mathbb S_{3d(d-2)}$ for $d\geq 4$ and in the case $d=3$ this group is the group of the projective transformations of $\mathbb P^2$ leaving invariant the nine inflection points of the Fermat curve of degree three.Comment: 19 page

On germs of finite morphisms of smooth surfaces

Questions related to deformations of germs of finite morphisms of smooth surfaces are discussed. A classification of the four-sheeted germs of finite covers $F: (U,o')\to (V,o)$ is given up to smooth deformations, where $(U,o')$ and $(V,o)$ are two connected germs of smooth complex-analytic surfaces. The singularity types of their branch curves and the local monodromy groups are investigated also.Comment: Definition of D-automorphisms and several misprints are correcte

On the variety of the inflection points of plane cubic curves

In the paper, we investigate properties of the nine-dimensional variety of the inflection points of the plane cubic curves. The description of local monodromy groups of the set of inflection points near singular cubic curves is given. Also, it is given a detailed description of the normalizations of the surfaces of the inflection points of plane cubic curves belonging to general two-dimensional linear systems of cubic curves, The vanishing of the irregularity a smooth manifold birationally isomorphic to the variety of the inflection points of the plane cubic curves is proved.Comment: 27 page

Factorizations in finite groups

A necessary condition for uniqueness of factorizations of elements of a finite group $G$ with factors belonging to a union of some conjugacy classes of $G$ is given. This condition is sufficient if the number of factors belonging to each conjugacy class is big enough. The result is applied to the problem on the number of irreducible components of the Hurwitz space of degree $d$ marked coverings of $\mathbb P^1$ with given Galois group $G$ and fixed collection of local monodromies.Comment: 29 page