algebraic variety

English

Noun

algebraic variety (plural algebraic varieties)

1. (algebraic geometry) The set of solutions of a given system of polynomial equations over the real or complex numbers; any of certain generalisations of such a set that preserves the geometric intuition implicit in the original definition.

   Synonym: (elliptical) variety

   • 2005, Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvili, Geometric and Algebraic Topological Methods in Quantum Mechanics, World Scientific, page 87:

     If ${\mathcal {W}}$ is a locally closed subset (i.e., the intersection of open and closed sets) of an algebraic variety, it becomes an algebraic variety in a natural manner since the germs of regular functions at $x\in {\mathcal {W}}$ axe taken to be the germs of functions on ${\mathcal {W}}$ induced by functions in the stalk ${\mathfrak {R}}_{x}$. The definitions of irreducibility and local rings of subvarieties for algebraic varieties are given in the same manner as before. From now on, by a variety is meant an algebraic variety. Any variety (X,${\mathfrak {R}})$, by definition, is a local-ringed space.

   • 2011, Carel Faber, Gerard van der Geer, Eduard Looijenga, Classification of Algebraic Varieties, European Mathematical Society, page vii:

     The quest for understanding the structure of algebraic varieties is a very natural one. […] A landmark from that time was Hironaka's 1964 theorem on the existence of a smooth model for any complex algebraic variety.

   • 2014, Herwig Hauser, “Blowups and Resolution”, in David Ellwood, Herwig Hauser, Shigefumi Mori, Josef Schicho, editors, The Resolution of Singular Algebraic Varieties, American Mathematical Society, page 7:

     Remark 2.8 Abstract algebraic varieties are obtained by gluing affine algebraic varieties along principal open subsets, cf. [Mum99] I, §3, §4, [Sha94] V, §3. This allows to develop the category of algebraic varieties with the usual constructions therein. All subsequent definitions could be formulated for abstract algebraic varieties, but will only be developed in the affine or quasi-affine case to keep things simple.

Usage notes

Algebraic varieties are the central objects of study in algebraic geometry.

Hyponyms

• (set of solutions to a system of polynomial equations; any of certain generalisations of such a set): affine variety, algebraic manifold, projective variety, algebraic curve

Translations

set of solutions to a system of polynomial equations; any of certain generalisations of such a set

• Finnish: varisto
• Galician: variedade alxebraica f
• Italian: varietà algebrica f
• Russian: алгебраи́ческое многообра́зие n (algebraíčeskoje mnogoobrázije)

See also

• algebraic set

Further reading

• Scheme (mathematics) on Wikipedia.Wikipedia
• algebraic variety on nLab