Related papers: Some open questions on arithmetic Zariski pairs
We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets.
Computing the real solutions to a system of polynomial equations is a challenging problem, particularly verifying that all solutions have been computed. We describe an approach that combines numerical algebraic geometry and sums of squares…
A central question in the study of line arrangements in the complex projective plane $\mathbb{CP}^2$ is: when does the combinatorial data of the arrangement determine its topological properties? In the present work, we introduce a…
In this article, examples of Zariski pairs $(B_1, B_2)$ satisfying the following condition are given: (i) $\deg B_1 = \deg B_2 = 7$. (ii) Irreducible components of $B_i$ $(i = 1, 2)$ are lines and conics. (iii) Singularities of $B_i$ $(i =…
Kaplansky conjectured that if two positive-definite real ternary quadratic forms have perfectly identical representations over $\mathbb{Z}$, they are constant multiples of regular forms, or is included in either of two families parametrized…
We compute Bernstein-Sato polynomials of some pairs of topologically equivalent plane curve singularities. Some pairs have the same Tjurina number but distinct Bernstein-Sato polynomials, which implies that they are not analytically…
For any pair of orientable closed hyperbolic $3$--manifolds, this paper shows that any isomorphism between the profinite completions of their fundamental groups witnesses a bijective correspondence between the Zariski dense…
We present a method to produce examples of non-homeomorphic conjugate complex varieties based on the genus theory of lattices. As an application, we give examples of arithmetic Zariski pairs.
In this work, we study a family of Cremona transformations of weighted projective planes which generalize the standard Cremona transformation of the projective plane. Starting from special plane projective curves we construct families of…
Empirical properties of generating systems for complex reflection groups and their braid groups have been observed by Orlik-Solomon and Brou\'e-Malle-Rouquier, using Shephard-Todd classification. We give a general existence result for…
Let $X$ be a smooth compact manifold. We propose a geometric model for the group $K^0(X,\mathbb{R}/\mathbb{Z}).$ We study a well-defined and non-degenerate analytic duality pairing between $K^0(X,\mathbb{R}/\mathbb{Z})$ and its Pontryagin…
Braided doubles provide a unifying framework for classical and quantum universal enveloping algebras and rational Cherednik algebras. They are a class of algebras with triangular decomposition, arising from a deformation problem, the…
We study, for plane complex branches of genus one, the topological type of its generic polar curve, as a function of the semigroup of values and the Zariski invariant of the branch. We improve some results given by Casas-Alvero in 2023,…
We partially prove and partially disprove Oka's conjecture on the fundamental group/Alexander polynomial of an irreducible plane sextic. Among other results, we enumerate all irreducible sextics with simple singularities admitting dihedral…
We classify the Rauzy-Veech groups of all connected components of all strata of the moduli space of translation surfaces in absolute homology, showing, in particular, that they are commensurable to arithmetic lattices of symplectic groups.…
A new notion of Cartan pairs as a substitute of notion of vector fields in noncommutative geometry is proposed. The correspondence between Cartan pairs and differential calculi is established.
Equivariant tree models are statistical models used in the reconstruction of phylogenetic trees from genetic data. Here equivariant refers to a symmetry group imposed on the root distribution and on the transition matrices in the model. We…
We introduce a class of inverse monoids, called Tarski monoids, that can be regarded as non-commutative generalizations of the unique countable, atomless Boolean algebra. These inverse monoids are related to a class of etale topological…
The classical Zariski-van Kampen theorem gives a presentation of the fundamental group of the complement of a complex algebraic curve in $\mathbb{P}^2$. The first generalization of this theorem to singular (quasi-projective) varieties was…
The finite-degree Zariski (Z-) closure is a classical algebraic object, that has found a key place in several applications of the polynomial method in combinatorics. In this work, we characterize the finite-degree Z-closures of a subclass…