Algebraic curve

An algebraic curve is a one-dimensional algebraic variety defined by polynomial equations. It may be studied over fields such as the real or complex numbers.

Algebraic variety: A geometric space defined as the common zero set of polynomial equations, with algebraic structure. An algebraic curve is the one-dimensional case of this broader kind of space.
Plane algebraic curve: An algebraic curve embedded in a plane and defined by one polynomial equation in two coordinates. Plane curves are the most direct examples of algebraic curves described by equations.
Genus (mathematics): A numerical invariant measuring a curve's geometric complexity, defined topologically or algebraically. Genus distinguishes broad classes of smooth curves and constrains their maps.
Differential geometry: The study of smooth manifolds equipped with geometric structures such as metrics and connections. Its curves are often smooth paths, not necessarily solution sets of polynomial equations.
Polynomial: An expression built from variables and coefficients using addition, multiplication, and nonnegative integer powers. Polynomials supply the equations whose shared solutions form an algebraic curve.
Homogeneous polynomial: A polynomial whose monomials all have the same total degree. Homogeneous equations define projective curves without privileging an affine coordinate chart.
Riemann–Roch theorem: A theorem relating the dimensions of spaces of meromorphic functions to divisors and genus. It turns curve geometry into precise counts of functions with prescribed poles.
Transcendental curve: A curve that cannot be defined by polynomial equations over the relevant field. It separates algebraic curves from curves described only by non-polynomial relations.
Dimension (algebraic geometry): A measure of the number of independent parameters needed to describe a geometric space algebraically. Dimension one distinguishes curves from algebraic points and surfaces.
Singular point of an algebraic variety: A point where a variety fails to be locally smooth. Singular points mark where a curve's local geometry departs from that of a smooth line.

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