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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 the common case of a real algebraic curve, where k is the field of real numbers, an algebraic curve is a finite union of topological curves. When complex zeros are considered, one has a complex algebraic curve, which, from the topological point of view, is not a curve, but a surface, and is often called a Riemann surface. Although not being ...
The hyperelliptic curve defined by = (+) (+) has only finitely many rational points (such as the points (,) and (,)) by Faltings's theorem.. In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. [1]
The cruciform curve, or cross curve is a quartic plane curve given by the equation = where a and b are two parameters determining the shape of the curve. The cruciform curve is related by a standard quadratic transformation, x ↦ 1/x, y ↦ 1/y to the ellipse a 2 x 2 + b 2 y 2 = 1, and is therefore a rational plane algebraic curve of genus zero.
One of the most important properties of stable curves is the fact that they are local complete intersections. This implies that standard Serre-duality theory can be used. In particular, it can be shown that for every stable curve ω C / S ⊗ 3 {\displaystyle \omega _{C/S}^{\otimes 3}} is a relatively very-ample sheaf; it can be used to embed ...
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.
If the curve is non-singular the geometric genus and the arithmetic genus are equal, but if the curve is singular, with only ordinary singularities, the geometric genus is smaller. More precisely, an ordinary singularity of multiplicity r {\displaystyle r} decreases the genus by 1 2 r ( r − 1 ) {\displaystyle {\frac {1}{2}}r(r-1)} .
In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) C is determined by its Jacobian variety J(C), when the latter is given in the form of a principally polarized abelian variety.