Algebraic Varieties: Basic Notions
Identifieur interne : 003423 ( Istex/Corpus ); précédent : 003422; suivant : 003424Algebraic Varieties: Basic Notions
Auteurs : I. R. ShafarevichSource :
- Encyclopaedia of Mathematical Sciences [ 0938-0396 ] ; 1994.
Abstract
Abstract: The aim of this chapter is to give a precise meaning to the following words: an algebraic variety is an object which is defined locally by some polynomial equations. The main distinction between algebraic varieties and differentiable or complex analytic manifolds (see Bourbaki [1967–1971], Chirka [1985], or Lang [1962]) lies in the choice of the local models. In the differentiable and complex analytic cases, these are the open subsets of ”n or ℂn. A local model of an algebraic variety is a subset of the coordinate space which is given by polynomial equations. This makes sense only if we fix a ground field K, over which both the polynomials and the solutions will be considered. In order to simplify the algebraic aspect of the question as much as possible and to concentrate on geometry, we shall assume in the first three chapters that the field K is algebraically closed. In the present chapter we shall also examine the simplest notions from algebraic geometry that have direct analogues in the differentiable and analytic cases.
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DOI: 10.1007/978-3-642-57878-6_6
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A local model of an algebraic variety is a subset of the coordinate space which is given by polynomial equations. This makes sense only if we fix a ground field K, over which both the polynomials and the solutions will be considered. In order to simplify the algebraic aspect of the question as much as possible and to concentrate on geometry, we shall assume in the first three chapters that the field K is algebraically closed. In the present chapter we shall also examine the simplest notions from algebraic geometry that have direct analogues in the differentiable and analytic cases.</div></front></TEI><istex><corpusName>springer-ebooks</corpusName><author><json:item><name>I. R. 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The main distinction between algebraic varieties and differentiable or complex analytic manifolds (see Bourbaki [1967–1971], Chirka [1985], or Lang [1962]) lies in the choice of the local models. In the differentiable and complex analytic cases, these are the open subsets of ”<Superscript>n</Superscript> or ℂ<Superscript>n</Superscript>. A local model of an algebraic variety is a subset of the coordinate space which is given by polynomial equations. This makes sense only if we fix a ground field <Emphasis Type="Italic">K</Emphasis>, over which both the polynomials and the solutions will be considered. In order to simplify the algebraic aspect of the question as much as possible and to concentrate on geometry, we shall assume in the first three chapters that the field <Emphasis Type="Italic">K</Emphasis> is algebraically closed. 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