Related papers: Braid Monodromy Factorization and Diffeomorphism T…

We give a criterion when a planar tree-like curve, i.e. a generic immersed plane curve each double point of which cuts it into two disjoint parts, can be send by a diffeomorphism of the plane onto a curve with no inflection points. We also…

We show that branched coverings of surfaces of large enough genus arise as characteristic maps of braided surfaces that is, lift to embeddings in the product of the surface with $\mathbb R^2$. This result is nontrivial already for…

Using symplectic topology and the Radon transform, we prove that smooth 4-dimensional projective planes are diffeomorphic to $\mathbb{CP}^2$. We define the notion of a plane curve in a smooth projective plane, show that plane curves in high…

We generalize results of the paper math.AG/9803144, in which Chisini's conjecture on the unique reconstruction of f by the curve B is investigated. For this fibre products of generic coverings are studied. The main inequality bounding the…

The embeddings of complex plane projective curves in the plane are a cornerstone of the topological study of algebraic varieties. In this work, we deal with the local and global aspects of these embeddings, with a special attention to its…

We describe symmetries of the braid monodromy decomposition for a class of plane curves defined over reals including the real curves with no real points and proving new divisibility relations for Alexander invariants of such curves.

We present a short proof of the fact that two irreducible germs of plane analytic curves are isotopic if they are equisingular, without recourse to the structure of the associated knots.

Consider the scheme parametrizing non-constant morphisms from a fixed projective curve to a projective surface. There is a rational map between this scheme and the Chow variety of $1$-cycles on the surface. We prove that, if the curve is…

In this paper, we prove that each automorphism of the Torelli group of a surface is induced by a diffeomorphism of the surface, provided that the surface is a closed, connected, orientable surface of genus at least 3. This result was…

We generalize results by Wakabayashi and Orevkov about rational cuspidal curves on the projective plane to that on $\mathbb{Q}$-homology projective planes. It turns out that the result is exactly the same as the projective plane case under…

Let $C \s \pr^2$ be an irreducible plane curve whose dual $C^* \s \pr^{2*}$ is an immersed curve which is neither a conic nor a nodal cubic. The main result states that the Poincar\'e group $\pi_1(\pr^2 \se C)$ contains a free group with…

We show that $\mathbb{C}^2$ contains pairs of properly embedded, smooth complex curves that are isotopic through homeomorphisms but not diffeomorphisms of $\mathbb{C}^2$. The construction is based on realizing corks as branched covers of…

We investigate the local contribution of the braid monodromy factorization in the context of the links obtained by the closure of these braids. We consider plane curves which are arrangements of lines and conics as well as some algebraic…

Suppose that $C\subset\mathbb P^2$ is a general enough nodal plane curve of degree $>2$, $\nu\colon \hat C\to C$ is its normalization, and $\pi\colon \hat C\to\mathbb P^1$ is a finite morphism simply ramified over the same set of points as…

A formula for factorizations of the full twist in the braid group $Br_{2m}$ depending on any four factorizations of the full twist in $Br_{m}$ is given. Applying this formula, a symplectic 4-manifold $X$ and two isotopic generic coverings…

We study the discriminant of a degree 4 extension given by a deformed bidouble cover, i.e., by equations z^2= u + a w, w^2= v + bz. We first show that the discriminant surface is a quartic which is cuspidal on a twisted cubic, i.e.,is the…

Let S be a compact, oriented surface with negative Euler characteristic and let f be a homeomorphism of S that is isotopic to the identity. If there exists a periodic orbit with a non-zero rotation vector, then there exists a simple braid…

We completely classify all plane curves of degree at most 30 with a unique cuspidal (locally unibranch) singular point and rational normalization in terms of the Newton pairs parameterizing the cusp. We distinguish between prime and…

Let T -> S be a finite flat morphism of degree two between regular integral schemes of dimension at most two (and with 2 invertible), having regular branch divisor D. We establish a bijection between Azumaya quaternion algebras on T and…

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…