Related papers: Some open questions on arithmetic Zariski pairs
This paper provides a non-standard analogue of Bezout's theorem. This is acheived by showing that, in all characteristics, the notion of Zariski multiplicity coincides with intersection multiplicity when we consider the full families of…
We consider spaces of plane curves in the setting of algebraic geometry and of singularity theory. On one hand there are the complete linear systems, on the other we consider unfolding spaces of bivariate polynomials of Brieskorn-Pham type.…
We study complex plane projective sextic curves with simple singularities up to equisingular deformations. It is shown that two such curves are deformation equivalent if and only if the corresponding pairs are diffeomorphic. A way to…
In this note we give examples of Zariski's pairs $B_{1,m}, B_{2,m}$ ($m \in N$ and $m \geq 5$) of plane cuspidal curves such that (i) $B_{i,m}$ is the discriminant curve of a generic morphism $f_{i,m}:S_i \to P^2$, $i=1, 2$, (ii) $S_1$ and…
In this present paper, we study the splitting of nodal plane curves with respect to contact conics. We define the notion of splitting type of such curves and show that it can be used as an invariant to distinguish the embedded topology of…
We study the critical points of monomial functions over an algebraic subset of the probability simplex. The number of critical points on the Zariski closure is a topological invariant of that embedded projective variety, known as its…
We consider a Ziegler pair of plane arrangements, that is two plane arrangements $\mathcal{A}:f=0$ and $\mathcal{A}':f'=0$ in the projective space $\mathbb{P}^3$, such that the intersection lattices $L(\mathcal{A})$ and $L(\mathcal{A}')$…
In this paper we discuss some properties of fundamental groups and Alexander polynomials of plane curves. We discuss the relationship of the non-triviality of Alexander polynomials and the notion of (nearly) freeness for irreducible plane…
A compatible associative algebra is a vector space endowed with two associative multiplication operations that satisfy a natural compatibility condition. In this paper, we investigate and classify compatible pairs of associative algebras of…
The logarithmic connections studied in the paper are direct images of regular connections on line bundles over genus-2 double covers of the elliptic curve. We give an explicit parametrization of all such connections, determine their…
In this paper we study the embedded topology of reducible plane curves having a smooth irreducible component. In previous studies, the relation between the topology and certain torsion classes in the Picard group of degree zero of the…
Consider an integer associated with every subset of the set of columns of an $n\times k$ matrix. The collection of those matrices for which the rank of a union of columns is the predescribed integer for every subset, will be denoted by…
We address the question of finding algebraic properties that are respectively equivalent, for a morphism between algebraic varieties over an algebraically closed field of characteristic zero, to be an homeomorphism for the Zariski topology…
We study the S-integral points on the complement of a union of hyperplanes in projective space, where S is a finite set of places of a number field k. In the classical case where S consists of the set of archimedean places of k, we…
Recently, the authors have proved the finiteness of common zeros of two iterated rational maps under some compositional independence assumptions. In this article, we advance towards a question of Hsia and Tucker on a Zariski non-density of…
We study the embedded topology of certain conic-line arrangements of degree 7. Two new examples of Zariski pairs are given. Furthermore, we determine the number of connected components of the conic-line arrangements. We also calculate the…
For line arrangements in P^2 with nice combinatorics (in particular, for those which are nodal away the line at infinity), we prove that the combinatorics contains the same information as the fundamental group together with the meridianal…
This paper gives an extension of the classical Zariski-van Kampen theorem describing the fundamental groups of the complements of plane singular curves by generators and relations. It provides a procedure for computation of the first…
A real number $\beta>1$ is called an Yrrap (or Ito-Sadahiro) number if the corresponding negative $\beta$-transformation defined by $x\mapsto 1-\{\beta x\}$ for $x\in[0,1]$, where $\{y\}$ denotes the fraction part of $y\in\mathbb{R}$, has a…
Constructing lattice isomorphic line arrangements that are not lattice isotopic is a complex yet fundamental task. In this paper, we focus on such pairs but which are not Galois conjugated, referred to as nonarithmetic pairs. Splitting…