Yagmyrova M.
GEOMETRIC STRUCTURES RIEMANN SURFACES AND ALGEBRAIC CURVES *
Аннотация:
this article provides brief information about the geometric structures of Riemann Surfaces and algebraic curves
Ключевые слова:
theory, surface, complex, sphere, curve, degree, structure, intersection
УДК 1
Yagmyrova M.
Lecturer of the Department of Applied mathematics and Informatics
Oguzhan University of Engineering Technologies of Turkmenistan
(Ashgabat, Turkmenistan)
GEOMETRIC STRUCTURES RIEMANN
SURFACES AND ALGEBRAIC CURVES
Abstract: this article provides brief information about the geometric structures of Riemann Surfaces and algebraic curves.
Keywords: theory, surface, complex, sphere, curve, degree, structure, intersection.
Riemannian geometry is a branch of differential geometry whose main object of study is Riemannian manifolds, that is, smooth manifolds with an additional structure, a Riemannian metric, in other words, with the choice of a Euclidean metric on each tangent space, and this metric changes smoothly from point to point. Sometimes, especially often in mathematical physics, Riemannian geometry is also understood as the pseudo-Riemannian geometry of manifolds with a pseudo-Riemannian metric, for example, the space-time geometry of special and general relativity.
A Riemann surface is a mathematical object, the traditional name in complex analysis for a one-dimensional complex differentiable manifold.
Examples of Riemann surfaces are the complex plane and the Riemann sphere. The Riemann surface allows you to geometrically represent multi-valued functions of a complex variable in such a way that each of its points corresponds to one value of a multi-valued function, and with continuous movement along the surface, the function also changes continuously. The canonical form of a Riemann surface is a representation in the form of a flat cake with a certain number of holes.
An algebraic curve, or a plane algebraic curve, is, in the simplest case, the set of zeros of a polynomial of two variables. The degree or order of an algebraic curve is the degree of this polynomial.
For example, the unit circle given by the equation
x^{2}+y^{2}=1, is an algebraic curve of the second degree, since it coincides with the set of zeros of the polynomial.
x^{2}+y^{2}-1}.
Plane algebraic curves from the first to the eighth degree are respectively called straight lines, conics, cubes, quartics, pentics, sextics, septics and octics. Algebraic geometry also considers not only the real zeros of polynomials, but also the complex ones. Moreover, polynomials can be considered over arbitrary fields.
Also in algebraic geometry, more general algebraic curves are considered, which are not necessarily contained in two-dimensional, but in spaces with a large number of dimensions, and also in projective spaces.
It turns out that many properties of an algebraic curve do not depend on the choice of a particular embedding in some space, which leads to the following general definition of an algebraic curve.
An algebraic curve is an algebraic variety of dimension 1. In other words, an algebraic curve is an algebraic variety, each algebraic subvariety of which is a singleton.
A rational curve, also known as a unicursal curve, is a curve that is birationally equivalent to an affine line (or a projective line); in other words, a curve that admits a rational parametrization.
More specifically, a rational curve in n-dimensional space can be parametrized (except for a certain number of isolated "singular points") with n rational functions of a single parameter t.
Any conic section over the field of rational numbers containing at least one rational point is a rational curve[1]. It can be parametrized by drawing a straight line with an arbitrary slope t through a rational point and assigning to this t the second intersection point of the straight line and the conic (there cannot be more than two)
Rational curves (over an algebraically closed field) are exactly algebraic curves of genus 0 (see below), in this terminology elliptic curves are curves of genus 1 with a rational point. Any such curve can be represented as a cube without singularities.
An elliptic curve carries the structure of an Abelian group. The sum of three points on a cube is equal to zero if and only if these points are collinear.
REFERENCES:
Номер журнала Вестник науки №5 (62) том 1
Ссылка для цитирования:
Yagmyrova M. GEOMETRIC STRUCTURES RIEMANN SURFACES AND ALGEBRAIC CURVES // Вестник науки №5 (62) том 1. С. 360 - 362. 2023 г. ISSN 2712-8849 // Электронный ресурс: https://www.вестник-науки.рф/article/8040 (дата обращения: 11.05.2024 г.)
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