Projective variety

（重定向自Projective curve）

An elliptic curve is a smooth projective curve of genus one.

In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space P over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. If the condition of generating a prime ideal is removed, such a set is called a projective algebraic set. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of P. A Zariski open subvariety of a projective variety is called a quasi-projective variety.

单词	Projective curve
释义	Projective curve 英语百科 Projective variety （重定向自Projective curve） In algebraic geometry, a projective variety over an algebraically closed field k is a subset of some projective n-space P over k that is the zero-locus of some finite family of homogeneous polynomials of n + 1 variables with coefficients in k, that generate a prime ideal, the defining ideal of the variety. If the condition of generating a prime ideal is removed, such a set is called a projective algebraic set. Equivalently, an algebraic variety is projective if it can be embedded as a Zariski closed subvariety of P. A Zariski open subvariety of a projective variety is called a quasi-projective variety.
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