Related papers: Topology of plane algebraic curves: the algebraic …
We study plane algebraic curves defined over a field k of arbitrary characteristic as coverings of the the projective line and the problem of enumerating branched coverings of $\mathbb{P}^{1}$ by using combinatorial methods.
We propose some problems on the classification of toric manifolds from the viewpoint of topology and survey related results.
Constructions and exploration of plane algebraic curves has received a new push with the development of automated methods, whose algorithms are continuously improved and implemented in various software packages. We use them to explore the…
We prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.
Computational topology is an area that revisits topological problems from an algorithmic point of view, and develops topological tools for improved algorithms. We survey results in computational topology that are concerned with graphs drawn…
We consider ruled surfaces with finite multiplicity. We study behaviors of the striction curves and the singularities of the ruled surfaces. We also give geometric meanings of invariants related to the ruled surfaces.
The study of topological information of spatial objects has for a long time been a focus of research in disciplines like computational geometry, spatial reasoning, cognitive science, and robotics. While the majority of these researches…
We enumerate plane complex algebraic curves of a given degree with one singularity of any given topological type. Our approach is to compute the homology classes of the corresponding equisingular strata in the parameter spaces of plane…
We give a survey of results on the geometry of complex algebraic Q-acyclic surfaces, so-called 'Q-homology planes', including some recent results.
A projective algebraic surface which is homeomorphic to a ruled surface over a curve of genus $g\ge 1$ is itself a ruled surface over a curve of genus $g$. In this note, we prove the analogous result for projective algebraic manifolds of…
In this paper we collect the main properties of free curves in the complex projective plane and a lot of conjectures and open problems, both old and new. In the quest to understand the mystery of free curves, many tools were developed and…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
A quadric in $\R P^3$ cuts a curve of degree 6 on a cubic surface in $\R P^3$. The papers classifies the nonsingular curves cut in this way on non-singular cubic surfaces up to homeomorphism. Two issues new in the study related to the first…
The study of the topology of real algebraic varieties dates back to the work of Harnack, Klein and Hilbert in the 19th century; in particular, the isotopy type classification of real algebraic curves in real toric surfaces is a classical…
In this paper, we study the computation of curvatures at the singular points of algebraic curves and surfaces. The idea is to convert the problem to compute the curvatures of the corresponding regular parametric curves and surfaces, which…
In this paper we present algorithms for computing the topology of planar and space rational curves defined by a parametrization. The algorithms given here work directly with the parametrization of the curve, and do not require to compute or…
This is a glossary of notions and methods related with the topological theory of collections of affine planes, including braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complexes of…
Our purpose in this article is first, following [14], to find the topological upper limits of projections of secant planes to $C^{1}$ surfaces and the topological upper limits of projections of secant hyperplanes to $C^{1}$ hypersurfaces…
Our aim is to reprove the basic results of the theory of branches of plane algebraic curves over algebraically closed fields of arbitrary characteristic. We do not use the Hamburger-Noether expansions. Our basic tool is the logarithmic…
The paper surveys several results on the topology of the space of arcs of an algebraic variety and the Nash problem on the arc structure of singularities.