arXiv subjects
Publications and source records attributed to Vik. S. Kulikov.
The article examines a set of irreducible germs $f_P:U_P\to V_p$ of %finite generic morphisms $f:S\to\mathbb P^2$ to the projective plane whose branch curve germs $B_P\subset V_p$ have singularities equisingular deformation equivalent to singularities given by equations $x^{k_1}-y^{k_2}=0$ with coprime $k_1,k_2\in\mathbb N$.
A text of the talk given by the author on Conference on Algebra, Algebraic Geometry and Number Theory dedicated to the 100th anniversary of I.R. Shafarevich, Moscow, June 5 -- 9, 2023.
It is proved that the braid monodromy group $Γ_{P(z)}$ of a polynomial $P(z)\in\mathbb C[z]$, $°P(z)=n$, is the braid group $\text{Br}_n$ if the polynomial $P(z)$ has $n-1$ distinct critical values.
In \cite{K-rig}, a map $β:\mathcal R\to\mathcal{B}el$ from the set $\mathcal R$ of equivalence classes of rigid germs of finite morphisms branched in germs of curves having $ADE$ singularity types onto the set $\mathcal{B}el$ of rational Belyi pairs $f:\mathbb P^1\to\mathbb P^1$ considered up to the action of $PGL(2,\mathbb C)$ was defined. In this article, the inverse images of this map are investigated in terms of monodromies of Belyi pairs.
We prove that a germ of a finite morphism of smooth surfaces is rigid if the germ of its branch curve has one of $ADE$-singularity types and establish a correspondence between the set of rigid germs and the set of Belyi rational functions $f\in \bar{\mathbb Q}(z)$.
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.
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 $B$.
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.
We prove that for a generic Lefschetz pencil of plane curves of degree $d\geq 3$ there exists a curve $H$ (called the Hesse curve of the pencil) of degree $6(d-1)$ and genus $3(4d^2-13d+8)+1$, and such that: $(i)$ $H$ has $d^2$ singular points of multiplicity three at the base points of the pencil and $3(d-1)^2$ ordinary nodes at the singular points of the degenerate members of the pencil; $(ii)$ for each member of the pencil the intersection of $H$ with this fibre consists of the inflection points of this member and the base points of the pencil.
In \cite{Ku0}, the ambiguity index $a_{(G,O)}$ was introduced for each equipped finite group $(G,O)$. It is equal to the number of connected components of a Hurwitz space parametrizing coverings of a projective line with Galois group $G$ assuming that all local monodromies belong to conjugacy classes $O$ in $G$ and the number of branch points is greater than some constant. We prove in this article that the ambiguity index can be identified with the size of a generalization of so called Bogomolov multiplier (\cite{Kun1}, see also \cite{BO87}) and hence can be easily computed for many pairs $(G,O)$.
Let $C_1$ be an irreducible component of a reduced projective curve $C\subset \mathbb P^2$ defined over the field $\mathbb C$, $\mathrm{deg} C_1\geq 2$, and let $T$ be the set of lines $l\subset \mathbb P^2$ meeting $C$ transversally. In the article, we prove that for a line $l_0\in T$ and any two points $P_1,P_2\in C_1\cap l_0$ there is a loop $l_t\subset T$, $t\in [0,1]$, such that the movement of the line $l_0$ along the loop $l_t$ induces the transposition of the points $P_1$, $P_2$ and the identity permutation of the other points of $C\cap l_0$.
This article is a continuation of the article with the same title (see arXiv:1003.2953v1). Let {\rm $\text{HUR}_{d,t}^{G}(\mathbb P^1)$} be the Hurwitz space of degree $d$ coverings of the projective line $\mathbb P^1$ with Galois group $\mathcal S_d$ and having fixed monodromy type $t$ consisting of a collection of local monodromy types (that is, a collection of conjugacy classes of permutations $σ$ of the symmetric group $\mathcal S_d$ acting on the set $I_d=\{1,...,d\}$). We prove that if the type $t$ contains big enough number of local monodromies belonging to the conjugacy class $C$ of an odd permutation $σ$ which leaves fixed $f_C\geq 2$ elements of $I_d$, then the Hurwitz space {\rm $\text{HUR}_{d,t}^{\mathcal S_d}(\mathbb P^1)$} is irreducible.
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.
For any reduced curve $C\subset \mathbb P^2$, we define the notions of the number of its virtual cusps $c_v$ and the number of its virtual nodes $n_v$ which are non-negative, coincide respectively with the numbers of ordinary cusps and nodes in the case of cuspidal curves, and if $\hat C$ is the dual curve of an irreducible curve $C$ and $\hat n_v$ and $\hat c_v$ are the numbers of its virtual nodes and virtual cusps, then the integers $c_v$, $n_v$, $\hat c_v$, $\hat n_v$ satisfy Classical Plücker's formulae.
We introduce a natural structure of a semigroup (isomorphic to a factorization semigroup of the unity in the symmetric group) on the set of irreducible components of Hurwitz space of marked degree $d$ coverings of $\mathbb P^1$ of fixed ramification types. It is proved that this semigroup is finitely presented. The problem when collections of ramification types define uniquely the corresponding irreducible components of the Hurwitz space is investigated. In particular, the set of irreducible components of the Hurwitz space of three-sheeted coverings of the projective line is completely described.
We investigate the problem of existence of degenerations of surfaces in $\mathbb P^3$ with ordinary singularities into plane arrangements in general position.
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.
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 $B$.