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An algebraic curve in the Euclidean plane is the set of the points whose coordinates are the solutions of a bivariate polynomial equation p(x, y) = 0.This equation is often called the implicit equation of the curve, in contrast to the curves that are the graph of a function defining explicitly y as a function of x.
In mathematics, Weber's theorem, named after Heinrich Martin Weber, is a result on algebraic curves. It states the following. Consider two non-singular curves C and C ′ having the same genus g > 1. If there is a rational correspondence φ between C and C ′, then φ is a birational transformation.
Following this purely algebraic problem I would like to raise a question that, it seems to me, can be attacked by the same method of continuous coefficient changing, and whose answer is of similar importance to the topology of the families of curves defined by differential equations – that is the question of the upper bound and position of ...
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects.
A space curve is a curve for which is at least three-dimensional; a skew curve is a space curve which lies in no plane. These definitions of plane, space and skew curves apply also to real algebraic curves , although the above definition of a curve does not apply (a real algebraic curve may be disconnected ).
The study of moduli spaces of curves, maps and other geometric objects, sometimes via the theory of quantum cohomology. The study of quantum cohomology, Gromov–Witten invariants and mirror symmetry gave a significant progress in Clemens conjecture. Enumerative geometry is very closely tied to intersection theory. [1]
In mathematics, a theta characteristic of a non-singular algebraic curve C is a divisor class Θ such that 2Θ is the canonical class.In terms of holomorphic line bundles L on a connected compact Riemann surface, it is therefore L such that L 2 is the canonical bundle, here also equivalently the holomorphic cotangent bundle.
In mathematics, the Cayley–Bacharach theorem is a statement about cubic curves (plane curves of degree three) in the projective plane P 2. The original form states: Assume that two cubics C 1 and C 2 in the projective plane meet in nine (different) points, as they do in general over an algebraically closed field. Then every cubic that passes ...