Related papers: Topology of plane algebraic curves: the algebraic …
This note is a survey on the topology of hyperplane arrangements. We mainly focus on the relationship between topology and the real structure, such as adjacent relations of chambers and stratifications related to real structures.
We present the topological classification of real parts of real regular elliptic surfaces with a real section.
There are two purposes in this article. One is to present a criterion for the existence of a birational embedding into a projective plane with inner and outer Galois points for algebraic curves. Another is to classify plane curves of degree…
The quest for regular models of arithmetic surfaces allows different viewpoints and approaches: using valuations or a covering by charts. In this article, we sketch both approaches and then show in a concrete example, how surprisingly…
In this paper, we define a new type of ruled surface called ruled surface by using the alternative frame of a base curve. Then, we study its differential geometric properties such as striction line, distribution parameter, fundamental…
We develop an essentially algebraic method to study biharmonic curves into an implicit surface. Although our method is rather general, it is especially suitable to study curves into surfaces defined by a polynomial equation: in particular,…
We study the geometry, Hodge theory and derived category of cubic fourfolds containing several planes and their associated twisted K3 surfaces. We focus on the case of two planes intersecting along a line.
We apply contact homology to obtain new results in the problem of distinguishing immersed plane curves without dangerous self-tangencies.
A classification of 2-dimensional surfaces imbedded in spacetime is presented, according to the algebraic properties of their shape tensor. The classification has five levels, and provides among other things a refinement of the concepts of…
Two plane analytic branches are topologically equivalent if and only if they have the same multiplicity sequence. We show that having same semigroup is equivalent to having same multiplicity sequence, we calculate the semigroup from a…
A criterion for the existence of a birational embedding into a projective plane with three collinear Galois points for algebraic curves is presented. The extendability of an automorphism induced by a Galois point to a linear transformation…
The splitting number is effective to distinguish the embedded topology of plane curves, and it is not determined by the fundamental group of the complement of the plane curve. In this paper, we give a generalization of the splitting number,…
We consider various equivalence relations on the set of homotopy classes of curves on a hyperbolic surface based on topological, algebraic, and geometric structures. The purpose of this work is to determine the relationship between these…
In this paper we use Groebner bases theory in order to determine planarity of intersections of two algebraic surfaces in ${\bf R}^3$. We specially considered plane sections of certain type of conoid which has a cubic egg curve as one of the…
In the article, we exhibit a series of new examples of rigid plane curves, that is, curves, whose collection of singularities determines them almost uniquely up to a projective transformation of the plane.
The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We…
Given the implicit equation $F(x,y,t,s)$ of a family of algebraic plane curves depending on the parameters $t,s$, we provide an algorithm for studying the topology types arising in the family. For this purpose, the algorithm computes a…
We study rectangles inscribed in lines in the plane by parametrizing these rectangles in two ways, one involving slope and the other aspect ratio. This produces two paths, one that finds rectangles with specified slope and the other…
In this paper, we give a Zariski triple of the arrangements for a smooth quartic and its four bitangents. A key criterion to distinguish the topology of such curves is given by a matrix related to the height pairing of rational points…
We obtain the list of automorphism groups for smooth plane sextic curves over an algebraically closed field K of characteristic p=0 or p>21. Moreover, we assign to each group a geometrically complete family over K describing its…