Mathematical object studied in the field of algebraic geometry
This article is about algebraic varieties. For the term "variety of algebras", see Variety (universal algebra).
The twisted cubic is a projective algebraic variety.
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.[1]: 58
Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility.
The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined by the set of its roots (a geometric object) in the complex plane. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is a defining feature of algebraic geometry.
Many algebraic varieties are differentiable manifolds, but an algebraic variety may have singular points while a differentiable manifold cannot. Algebraic varieties can be characterized by their dimension. Algebraic varieties of dimension one are called algebraic curves and algebraic varieties of dimension two are called algebraic surfaces.
In the context of modern scheme theory, an algebraic variety over a field is an integral (irreducible and reduced) scheme over that field whose structure morphism is separated and of finite type.
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Algebraicvarieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraicvariety is defined as...
mathematics, an algebraic group is an algebraicvariety endowed with a group structure that is compatible with its structure as an algebraicvariety. Thus the...
In algebraic geometry, a morphism between algebraicvarieties is a function between the varieties that is given locally by polynomials. It is also called...
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems...
are purely algebraic and rely on commutative algebra. Some are restricted to algebraicvarieties while others apply also to any algebraic set. Some are...
In algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the...
In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space P n {\displaystyle \mathbb {P}...
in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraicvariety that is also an algebraic group...
In the mathematical field of algebraic geometry, a singular point of an algebraicvariety V is a point P that is 'special' (so, singular), in the geometric...
universal algebra, an algebraic structure is called an algebra; this term may be ambiguous, since, in other contexts, an algebra is an algebraic structure...
In algebraic geometry, a complex algebraicvariety is an algebraicvariety (in the scheme sense or otherwise) over the field of complex numbers. Chow's...
mathematics, algebraic geometry and analytic geometry are two closely related subjects. While algebraic geometry studies algebraicvarieties, analytic geometry...
In algebraic geometry, the function field of an algebraicvariety V consists of objects that are interpreted as rational functions on V. In classical algebraic...
generally, an algebraic curve is an algebraicvariety of dimension one. (In some contexts, an algebraic set of dimension one is also called an algebraic curve...
element of the system Variety (universal algebra), classes of algebraic structures defined by equations in universal algebraVariety (linguistics), a specific...
commutative algebra, a major area of modern mathematics. Because these three fields (algebraic geometry, algebraic number theory and commutative algebra) are...
In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A...
In mathematics, a rational variety is an algebraicvariety, over a given field K, which is birationally equivalent to a projective space of some dimension...
Over an algebraically closed field, there is a stronger result about algebraic groups as algebraicvarieties: every connected linear algebraic group over...
polynomial algebras are used for the study of algebraicvarieties, which are solution sets of systems of polynomial equations. Weyl algebras and Lie algebras may...
ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings...
In algebraic geometry, a toric variety or torus embedding is an algebraicvariety containing an algebraic torus as an open dense subset, such that the...
stack quotients). For example, noncommutative algebraic geometry is supposed to extend a notion of an algebraic scheme by suitable gluing of spectra of noncommutative...
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory...
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