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- In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety X amounts to understanding the different ways of mapping X into projective space. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor. In more detail, a line bundle is called basepoint-free if it has enough sections to give a morphism to projective space. A line bundle is semi-ample if some positive power of it is basepoint-free; semi-ampleness is a kind of "nonnegativity". More strongly, a line bundle on X is very ample if it has enough sections to give a closed immersion (or "embedding") of X into projective space. A line bundle is ample if some positive power is very ample. An ample line bundle on a projective variety X has positive degree on every curve in X. The converse is not quite true, but there are corrected versions of the converse, the Nakai–Moishezon and Kleiman criteria for ampleness. (en)
- 대수기하학에서 풍부한 가역층(豐富한可逆層, 영어: ample invertible sheaf)은 그 거듭제곱의 단면들을 사영 공간의 동차 좌표로 간주하여 대수다양체를 사영 공간에 매장시킬 수 있는 가역층이다. 복소수체 위에서, 이는 가역층의 천 특성류가 켈러 구조로 표현됨을 뜻한다. (ko)
- 代数幾何学では、非常に豊富な直線束(very ample line bundle)は、基礎となる代数多様体や多様体 M から射影空間への埋め込みを行う設定に充分な大域的切断があるバンドルのことを言う。豊富な直線束(ample line bundle)はバンドルのある正のべきが非常に豊富となるときを言う。大域的に生成された層(globally generated sheaves)とは、射影空間への射を定義することに充分な切断を持つ層のことを言う。 (ja)
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- 대수기하학에서 풍부한 가역층(豐富한可逆層, 영어: ample invertible sheaf)은 그 거듭제곱의 단면들을 사영 공간의 동차 좌표로 간주하여 대수다양체를 사영 공간에 매장시킬 수 있는 가역층이다. 복소수체 위에서, 이는 가역층의 천 특성류가 켈러 구조로 표현됨을 뜻한다. (ko)
- 代数幾何学では、非常に豊富な直線束(very ample line bundle)は、基礎となる代数多様体や多様体 M から射影空間への埋め込みを行う設定に充分な大域的切断があるバンドルのことを言う。豊富な直線束(ample line bundle)はバンドルのある正のべきが非常に豊富となるときを言う。大域的に生成された層(globally generated sheaves)とは、射影空間への射を定義することに充分な切断を持つ層のことを言う。 (ja)
- In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety X amounts to understanding the different ways of mapping X into projective space. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor. (en)
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