In this paper, an algorithm to determine the topology of an algebraic space curve and to compute a certified G rational parametric approximation of the algebraic space curve is given by extending the local generic position method for solving zero dimensional polynomial equation systems to the case of dimension one. By certified, we mean the approximation curve and the original curve have the sa...

We begin this paper with a discussion of n-dimensional algebraic Vector bundles. Vector bundles of rank n may be identified with locally free Ox-modules of rank n. Several sections are dedicated to reviewing the basic notions of scheme-theory. We show how to construct the projectization of arbitrary rings. This construction corresponds to the construction of the tautological line bundle over an...

We use methods of algebraic topology to study the eigenvalue structure of a complex Osserman algebraic curvature tensor. We classify the algebraic curvature tensors which are both Osserman and complex Osserman in all but a finite number of exceptional dimensions.

An MI-group is an algebraic structure based on a generalization of the concept of a monoid that satisfies the cancellation laws and is endowed with an invertible anti-automorphism representing inversion. In this paper, a topology is defined on an MI-group $G$ under which $G$ is  a topological MI-group. Then we will identify open, discrete and compact MI-subgroups. The connected components of th...

As topology-like mathematical structures, convex structures can be characterized by betweenness relations via (restricted) hull operators in spaces. In a topological approach, the aim of this paper is to present fuzzy counterpart based on points Concretely, notion L-betweenness restricted L-hull introduced. Firstly, it proved that are categorically isomorphic and L-remotehood systems, respectiv...

The concept of a finitely related algebra, as opposed to the ones of finitely presentable and finitely generated ones, is not preserved under categorical equivalences. We propose a categorically well behaved approximation for it in the context of locally presentable categories, which turns out to be a natural counterpart to the (slightly reformulated) categorical definitions of finitely present...

We ask whether the notion of a homotopy class of a path on a complex algebraic variety admits a purely algebraic characterisation, and reformulate this question as a question of categoricity of the universal covering space of a complex algebraic variety in a natural countable in nitary language. We provide partial positive results towards the question. Assuming a conjecture of Shafarevich and s...

We discuss the principle tools and results and state a few open problems concerning the classification and topology of plane sextics and trigonal curves in ruled surfaces.

Anabelian geometry is a grand synthesis of number theory, algebraic geometry, algebraic topology, and group theory. Parallel to the situation in algebraic topology where to a space, one attaches a fundamental group (after fixing a basepoint), in algebraic geometry one attaches to every algebraic variety X (and a geometric point x) its étale fundamental group πét 1 (X,x). Similar to the topologi...

HiX and H X = ⊕ H iX. We refer to [Spa66] for the details and basic properties of these constructions, summarizing the most relevant facts below. One sacrifices some geometric intuition in working with cohomology instead of homology, but one gains the advantage of an easily defined ring structure. If σ ∈ CkX is a singular simplex, let fiσ ∈ CiX be the restriction of σ to the front i-face of the...