- adj.

- 豊富な、十分な

- 関
- abundance、abundant、abundantly、adequate、adequately、affluent、enough、full、fully、plentiful、rich、richly、satisfactorily、sufficient、sufficiently

- more than enough in size or scope or capacity; "had ample food for the party"; "an ample supply"
- affording an abundant supply; "had ample food for the party"; "copious provisions"; "food is plentiful"; "a plenteous grape harvest"; "a rich supply" (同)copious, plenteous, plentiful, rich
- fairly large; "a sizable fortune"; "an ample waistline"; "of ample proportions" (同)sizable, sizeable
- make (a garment) fuller by pleating or gathering
- having the normally expected amount; "gives full measure"; "gives good measure"; "a good mile from here" (同)good
- filled to satisfaction with food or drink; "a full stomach" (同)replete
- complete in extent or degree and in every particular; "a full game"; "a total eclipse"; "a total disaster" (同)total
- (of sound) having marked deepness and body; "full tones"; "a full voice"
- beat for the purpose of cleaning and thickening; "full the cloth"
- containing as much or as many as is possible or normal; "a full glass"; "a sky full of stars"; "a full life"; "the auditorium was full to overflowing"
- very productive; "rich seams of coal"
- containing plenty of fat, or eggs, or sugar; "rich desserts"; "they kept gorging on rich foods"
- having an abundant supply of desirable qualities or substances (especially natural resources); "blessed with a land rich in minerals"; "rich in ideas"; "rich with cultural interest"
- high in mineral content; having a high proportion of fuel to air; "a rich vein of copper", "a rich gas mixture"
- of great worth or quality; "a rich collection of antiques"
- pleasantly full and mellow; "a rich tenor voice"
- possessing material wealth; "her father is extremely rich"; "many fond hopes are pinned on rich uncles"
- suggestive of or characterized by great expense; "a rich display"
- about average; acceptable; "more than adequate as a secretary" (同)passable, fair to middling, tolerable
- sufficient for the purpose; "an adequate income"; "the food was adequate"; "a decent wage"; "enough food"; "food enough" (同)decent, enough
- having the requisite qualities or resources to meet a task; "she had adequate training"; "her training was adequate"; "she was adequate to the job"; "he was equal to the task" (同)equal
- the property of a more than adequate quantity or supply; "an age of abundance" (同)copiousness, teemingness
- (chemistry) the ratio of the total mass of an element in the earth
*s crust to the total mass of the earth*s crust; expressed as a percentage or in parts per million - (physics) the ratio of the number of atoms of a specific isotope of an element to the total number of isotopes present
- as much as necessary; "Have I eaten enough?"; (`plenty
*is nonstandard) "I*ve had plenty, thanks" (同)plenty - an adequate quantity; a quantity that is large enough to achieve a purpose; "enough is as good as a feast"; "there is more than a sufficiency of lawyers in this country" (同)sufficiency
- injure by trampling or as if by trampling; "The passerby was trampled by an elephant"
- the sound of heavy treading or stomping; "he heard the trample of many feet" (同)trampling
- the population is divided into strata and a random sample is taken from each stratum (同)representative sample, proportional sample
- items selected at random from a population and used to test hypotheses about the population (同)sample, sampling
- someone who injures by trampling

- 広い,広々とした / 《数えられない名詞・複数名詞の前に用いて》『十分な』(enough),あり余るほどたくさんの(abundant) / ふくよかな,豊満な(full)
- 『いっぱいの』,満ちた / (数量が)『たくさんある』,たっぷりある / (程度・量・数などが)『完全な』 / 《名詞の前にのみ用いて》最大限の / 《名詞の前にのみ用いて》正式の,本格的な / 《補語にのみ用いて》(…で)(腹・胸・頭が)いっぱいの(で)《+『of』+『名』》 / (形・体の一部が)ふっくらした,盛り上がった / (衣服などが)ゆったりしている / 《名詞の前にのみ用いて》(声・におい・色が)豊かな,濃い / まともに / 非常に(very) / 全部;十分;絶頂
- 『金持ちの』,『富んだ』,豊かな / 《名詞的に》《the~;複数扱い》金持ちの人たち / (衣服・宝石・家具などが)高価でみごとな,ぜいたくな / (想像力などが)豊かな;(土地が)肥えた / (音・声が)豊かな,深みのある;(色が)濃い,あざやかな;(料理・酒の問が)濃くておいしい;(香りが)強い / 濃度が高い;(特に食物の)栄養価が高い,(油・卵・バターなどを多く含んで)濃厚な / 《話》非常におもしろい,こっけいな;《反晴的に》とんでもない,ばかげた
- (要求・目的に)『十分な』 / かろうじて満足できる / 《補語にのみ用いて》(…に)『適した』,かなった《+『to』(『for』)+『名』》
- (あり余るほど)『たくさん』,『豊富』;《an~》たくさん(の…)《+『of』+『名』》
- 『不足のない』,十分な / 『十分な量(数)』,足りる量 / 『十分に』(sufficiently) / 全く,すっかり / もうたくさんだ,やめてくれ(Stop!)
- …‘を'『踏みつける』,踏みつぶす《+down+名,+名+down》 / (…を)『踏みつける』,踏みにじる《+on(upon,over)+名》 / 踏みつけること,踏みつける音
- 前例のない,空前の

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2016/03/01 17:47:13」(JST)

In algebraic geometry, a **very ample line bundle** is one with enough global sections to set up an embedding of its base variety or manifold into projective space. An **ample line bundle** is one such that some positive power is very ample. **Globally generated sheaves** are those with enough sections to define a morphism to projective space.

- 1 Introduction
- 1.1 Inverse image of line bundle and hyperplane divisors
- 1.2 Sheaves generated by their global sections
- 1.3 Very ample line bundles

- 2 Definitions
- 3 Criteria for ampleness of line bundles
- 3.1 Intersection theory
- 3.2 Sheaf cohomology

- 4 Generalizations
- 4.1 Vector bundles of higher rank
- 4.2 Big line bundles

- 5 See also
- 5.1 General algebraic geometry
- 5.2 Ampleness in complex geometry

- 6 References
- 6.1 Study references
- 6.2 Research texts

Given a morphism , any vector bundle on *Y*, or more generally any sheaf in modules, *e.g.* a coherent sheaf, can be pulled back to *X*, (see Inverse image functor). This construction preserves the condition of being a line bundle, and more generally the rank.

The notions described in this article are related to this construction in the case of morphisms to projective spaces

- and ,

the line bundle corresponding to the hyperplane divisor, whose sections are the 1-homogeneous regular functions. See Algebraic geometry of projective spaces#Divisors and twisting sheaves.

Let *X* be a scheme or a complex manifold and *F* a sheaf on *X*. One says that *F* is **generated by (finitely many) global sections** , if every stalk of *F* is generated as a module over the stalk of the structure sheaf by the germs of the *a _{i}*. For example, if

such that the pullback *f**(*O*(1)) is *F* (Note that this evaluation makes sense when *F* is a subsheaf of the constant sheaf of rational functions on *X*). The converse statement is also true: given such a morphism *f*, the pullback of *O*(1) is generated by its global sections (on *X*).

More generally, a **sheaf generated by global sections** is a sheaf *F* on a locally ringed space *X*, with structure sheaf *O*_{X} that is of a rather simple type. Assume *F* is a sheaf of abelian groups. Then it is asserted that if *A* is the abelian group of global sections, i.e.

then for any open set *U* of *X*, ρ(*A*) spans *F*(*U*) as an *O*_{U}-module. Here

is the restriction map. In words, all sections of *F* are locally generated by the global sections.

An example of such a sheaf is that associated in algebraic geometry to an *R*-module *M*, *R* being any commutative ring, on the spectrum of a ring *Spec*(*R*). Another example: according to Cartan's theorem A, any coherent sheaf on a Stein manifold is spanned by global sections.

Given a scheme *X* over a base scheme *S* or a complex manifold, a line bundle (or in other words an invertible sheaf, that is, a locally free sheaf of rank one) *L* on *X* is said to be **very ample**, if there is an embedding *i : X →* **P**^{n}_{S}, the *n*-dimensional projective space over *S* for some *n*, such that the pullback of the standard twisting sheaf *O*(1) on **P**^{n}_{S} is isomorphic to *L*:

Hence this notion is a special case of the previous one, namely a line bundle is very ample if it is globally generated and the morphism given by some global generators is an embedding.

Given a very ample sheaf *L* on *X* and a coherent sheaf *F*, a theorem of Serre shows that (the coherent sheaf) *F ⊗ L ^{⊗n}* is generated by finitely many global sections for sufficiently large

The notion of **ample line bundles** *L* is slightly weaker than very ample line bundles: a line bundle *L* is ample if for any coherent sheaf *F* on *X*, there exists an integer *n(F)*, such that *F* ⊗ *L*^{⊗n} is generated by its global sections for *n > n(F)*.

An equivalent, maybe more intuitive, definition of the ampleness of the line bundle is its having a positive tensorial power that is very ample. In other words, for there exists a projective embedding such that , that is the zero divisors of global sections of are hyperplane sections.

This definition makes sense for the underlying *divisors* (Cartier divisors) ; an ample is one where *moves in a large enough linear system*. Such divisors form a cone in all divisors of those that are, in some sense, *positive enough*. The relationship with projective space is that the for a very ample corresponds to the hyperplane sections (intersection with some hyperplane) of the embedded .

The equivalence between the two definitions is credited to Jean-Pierre Serre in Faisceaux algébriques cohérents.

Further information: intersection theory § Intersection theory in algebraic geometry

To decide in practice when a Cartier divisor *D* corresponds to an ample line bundle, there are some geometric criteria.

For curves, a divisor *D* is very ample if and only if *l*(*D*) = 2 + *l*(*D* − *A* − *B*) whenever *A* and *B* are points. By the Riemann–Roch theorem every divisor of degree at least 2*g* + 1 satisfies this condition so is very ample. This implies that a divisor is ample if and only if it has positive degree. The canonical divisor of degree 2*g* − 2 is very ample if and only if the curve is not a hyperelliptic curve.

The **Nakai–Moishezon criterion** (Nakai 1963, Moishezon 1964) states that a Cartier divisor *D* on a proper scheme *X* over an algebraically closed field is ample if and only if *D*^{dim(Y)}.*Y* > 0 for every closed integral subscheme *Y* of *X*. In the special case of curves this says that a divisor is ample if and only if it has positive degree, and for a smooth projective algebraic surface *S*, the Nakai–Moishezon criterion states that *D* is ample if and only if its self-intersection number *D*.*D* is strictly positive, and for any irreducible curve *C* on *S* we have *D*.*C* > 0.

The **Kleiman condition** states that for any projective scheme *X*, a divisor *D* on *X* is ample if and only if *D*.*C* > 0 for any nonzero element *C* in the closure of NE(*X*), the cone of curves of *X*. In other words a divisor is ample if and only if it is in the interior of the real cone generated by nef divisors.

Nagata (1959) constructed divisors on surfaces that have positive intersection with every curve, but are not ample. This shows that the condition *D*.*D* > 0 cannot be omitted in the Nakai–Moishezon criterion, and it is necessary to use the closure of NE(*X*) rather than NE(*X*) in the Kleiman condition.

Seshadri (1972, Remark 7.1, p. 549) showed that a line bundle *L* on a complete algebraic scheme is ample if and only if there is some positive ε such that deg(*L*|_{C}) ≥ ε*m*(*C*) for all integral curves *C* in *X*, where *m*(*C*) is the maximum of the multiplicities at the points of *C*.

The theorem of Cartan-Serre-Grothendieck states that for a line bundle on a variety , the following conditions are equivalent:

- is ample
- for
*m*big enough, is very ample - for any coherent sheaf on
*X*, the sheaf is generated by global sections, for*m*big enough

If is proper over some noetherian ring, this is also equivalent to:

- for any coherent sheaf on
*X*, the higher cohomology groups vanish for*m*big enough.

A locally free sheaf (vector bundle) on a variety is called **ample** if the invertible sheaf on is ample Hartshorne (1966).

Ample vector bundles inherit many of the properties of ample line bundles.

Main article: Iitaka dimension

An important generalization, notably in birational geometry, is that of a **big line bundle**. A line bundle on *X* is said to be big if the equivalent following conditions are satisfied:

- is the tensor product of an ample line bundle and an effective line bundle
- the Hilbert polynomial of the finitely generated graded ring has degree the dimension of
*X* - the rational mapping of the total system of divisors is birational on its image for .

The interest of this notion is its stability with respect to rational transformations.

- Cartier divisor
- Algebraic geometry of projective spaces
- Fano variety: a variety whose canonical line bundle is anti-ample

- Holomorphic vector bundle
- A line bundle is ample if and only if its Chern class is a Kähler class.
- Kodaira embedding theorem: for compact complex manifolds, ampleness and positivity coincide.
- Lefschetz hyperplane theorem: the study of very ample line bundles on complex projective manifolds gives strong topological information

- Hartshorne, Robin (1977),
*Algebraic Geometry*, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157 - Lazarsfeld, Robert (2004),
*Positivity in Algebraic Geometry*, Berlin: Springer-Verlag - The slides on ampleness in Vladimir Lazić's Lectures on algebraic geometry

- Hartshorne, Robin (1966), "Ample vector bundles",
*Publications Mathématiques de l'IHÉS*(29): 63–94, ISSN 1618-1913, MR 0193092 - Kleiman, Steven L. (1966), "Toward a numerical theory of ampleness",
*Annals of Mathematics. Second Series*(Annals of Mathematics)**84**(3): 293–344, doi:10.2307/1970447, ISSN 0003-486X, JSTOR 1970447, MR 0206009 - Moishezon, B. G. (1964), "A projectivity criterion of complete algebraic abstract varieties",
*Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya***28**: 179–224, ISSN 0373-2436, MR 0160782 - Nagata, Masayoshi (1959), "On the 14th problem of Hilbert",
*American Journal of Mathematics*(The Johns Hopkins University Press)**81**(3): 766–772, doi:10.2307/2372927, JSTOR 2372927, MR 0154867 - Nakai, Yoshikazu (1963), "A criterion of an ample sheaf on a projective scheme",
*American Journal of Mathematics*(The Johns Hopkins University Press)**85**(1): 14–26, doi:10.2307/2373180, ISSN 0002-9327, JSTOR 2373180, MR 0151461 - Seshadri, C. S. (1972), "Quotient spaces modulo reductive algebraic groups",
*Annals of Mathematics. Second Series*(Annals of Mathematics)**95**(3): 511–556, doi:10.2307/1970870, ISSN 0003-486X, JSTOR 1970870, MR 0309940

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リンク元 | 「adequate」「adequately」「satisfactorily」「full」「十分」 |

拡張検索 | 「autosampler」「Wilcoxon two-sample test」「blood sample」 |

- adj.

- 適切な、適当な、妥当な、十分な、適した

- 関
- adequately、ample、appropriate、appropriately、apt、correct、enough、fit、fittingly、full、fully、germane、neatly、pertinent、proper、properly、reasonable、richly、satisfactorily、sufficient、sufficiently、suitable、suitably、suited、valid

- adv.

- 適切に、適当に、十分に

- 関
- adequate、ample、appropriate、appropriately、apt、enough、fit、fittingly、full、fully、neatly、pertinent、proper、properly、reasonable、richly、satisfactorily、sufficient、sufficiently、suitably

- adv.

- 満足に、良好に、十分に、順調に

- 関
- adequate、adequately、ample、enough、excellent、favorable、favorably、full、fully、pleasure、richly、satisfaction、satisfactory、satisfy、sufficient、sufficiently、well

- adj.

- 完全な、満ちた、十分な

- 関
- adequate、adequately、ample、complete、completely、enough、entirely、fully、integral、perfect、perfectly、replete、richly、satisfactorily、sufficient、sufficiently

- 英
- sufficient、full、adequate、enough、ample、sufficiently、fully、enough、adequately、satisfactorily、richly
- 関
- 完全、妥当、適切、適当、豊富、満足、良好、適した、満ちた、順調