Meaning of algebraic variety | Babel Free

Definitions

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.

Equivalents

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.”

“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.”

“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.”

CEFR level