Algebraic variety

Algebraic varieties are the central objects of study in <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, a sub-field of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>. Classically, an algebraic variety is defined as the <a href="/facts/Solution_set/48N9PUjI">set of solutions</a> of a <a href="/facts/System_of_polynomial_equations/1456MrT2">system of polynomial equations</a> over the <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.: 58  
Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be <a href="/facts/Irreducible_component/MxWWaCM7">irreducible</a>, which means that it is not the <a href="/facts/Union_(set_theory)/KVVXKfWl">union</a> of two smaller <a href="/facts/Set_(mathematics)/BfucfHMq">sets</a> that are <a href="/facts/Closed_set/upYnRTUj">closed</a> in the <a href="/facts/Zariski_topology/uolkKBpM">Zariski topology</a>. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility.
The <a href="/facts/Fundamental_theorem_of_algebra/vspflSsM">fundamental theorem of algebra</a> establishes a link between <a href="/facts/Algebra/nMdqczjo">algebra</a> and <a href="/facts/Geometry/wTQf5ffQ">geometry</a> by showing that a <a href="/facts/Monic_polynomial/4JzuIJ5R">monic polynomial</a> (an algebraic object) in one variable with complex number coefficients is determined by the set of its <a href="/facts/Zero_of_a_function/mlK3dhoT">roots</a> (a geometric object) in the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>. Generalizing this result, <a href="/facts/Hilbert%27s_Nullstellensatz/TlgoRHuM">Hilbert's Nullstellensatz</a> provides a fundamental correspondence between <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideals</a> of <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial rings</a> and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of <a href="/facts/Ring_theory/VW6svEFe">ring theory</a>. This correspondence is a defining feature of algebraic geometry.
Many algebraic varieties are <a href="/facts/Differentiable_manifold/aSnbXKcV">differentiable manifolds</a>, but an algebraic variety may have <a href="/facts/Singular_point_of_an_algebraic_variety/8nvkrkE9">singular points</a> while a differentiable manifold cannot. Algebraic varieties can be characterized by their <a href="/facts/Dimension_of_an_algebraic_variety/sXqIHcYc">dimension</a>. Algebraic varieties of dimension one are called <a href="/facts/Algebraic_curve/LnWsPmDP">algebraic curves</a> and algebraic varieties of dimension two are called <a href="/facts/Algebraic_surface/2zHU0o9J">algebraic surfaces</a>.
In the context of modern <a href="/facts/Scheme_(mathematics)/c7uSqGqL">scheme</a> theory, an algebraic variety over a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> is an integral (irreducible and reduced) scheme over that field whose <a href="/facts/Structure_morphism/vBztnQUK">structure morphism</a> is separated and of finite type.

Algebraic variety open-in-new

Algebraic variety