algebraic variety

plural algebraic varieties

1. algebraic geometryThe 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.

   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.

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

   2 more examples

   If #92;mathcal#123;W#125; 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#92;in#92;mathcal#123;W#125; axe taken to be the germs of functions on #92;mathcal#123;W#125; induced by functions in the stalk #92;mathfrak#123;R#125;#95;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,#92;mathfrak#123;R#125;), by definition, is a local-ringed space.

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

   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.

   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

   Synonyms: variety