# Dictionary Definition

ample adj

1 more than enough in size or scope or capacity;
"had ample food for the party"; "an ample supply" [ant: meager]

2 affording an abundant supply; "had ample food
for the party"; "copious provisions"; "food is plentiful"; "a
plenteous grape harvest"; "a rich supply" [syn: copious, plenteous, plentiful, rich]

3 fairly large; "a sizable fortune"; "an ample
waistline"; "of ample proportions" [syn: sizable, sizeable]

# User Contributed Dictionary

## English

### Etymology

From ample, from amplus, probably for ambiplus, the last syllable akin to Latin plenus.### Pronunciation

- US: /ˈæmpl/
- Rhymes: -æmpəl

### Adjective

- Large; great in
size, extent, capacity, or bulk; spacious; roomy; widely extended.
- All the people in that ample house Did to that image bow their humble knees. --Spenser.

- Fully sufficient; abundant; liberal; copious; as, an ample fortune; ample justice.
- Not contracted of brief; not concise; extended; diffusive; as, an ample narrative.

#### Synonyms

# Extensive Definition

In mathematics, in algebraic
geometry or the theory of complex
manifolds, a very ample line bundle L
is one with enough sections
to set up an embedding
of its base variety
or manifold M into projective
space. The notions of ample line bundles and globally generated
sheaves are precursors of very ample line bundles.

## Sheaves generated by their global sections

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 a_i \in F(X), if every stalk of F is generated by the germs of the ai. For example, if F happens to be a line bundle, i.e. locally free of rank 1, this amounts to having finitely many global sections, such that for any point x in X, there is at least one section not vanishing at this point. In this case a choice of such global generators a0, ..., an gives a morphism- f: X → Pn, x ↦ [a0(x): ... : an(x)],

## Very ample line bundles

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 immersion i : X → PnS, the n-dimensional projective space over S for some n, such that the pullback of the standard twisting sheaf O(1) on PnS is isomorphic to L:- i*(O(1)) ≅ 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
immersion.

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 n. This in turn implies that global sections and higher
(Zariski) cohomology
groups

- Hi(X, F)

## Ample line bundles

The notion of ample line bundles L is slightly weaker than very ample line bundles: L is called ample if some tensor power L⊗n is very ample. This is equivalent to the following definition: 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.These definitions make sense for the underlying
divisors (Cartier
divisors) D; an ample D is one for which nD moves in a large
enough
linear system. Such divisors form a cone in
all divisors, of those which are in some sense positive enough''.
The relationship with projective space is that the D for a very
ample L will correspond to the hyperplane
sections (intersection with some hyperplane) of the embedded
M.

There is a more general theory of ample vector
bundles.

## Criteria for ampleness

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

For example. for a smooth algebraic
surface S, the Nakai-Moishezon criterion states that D is ample
iff its self-intersection
number is strictly positive, and for any irreducible curve C on
S we have

- D.C > 0

in the sense of
intersection theory.

Another useful criterion is the Kleiman
condition. This states that for any complete algebraic scheme X, a
divisor D on X is ample iff D.x > 0 for any nonzero element x in
the closure of NE(X),
the cone of curves of X. (Note that taking the closure is necessary
here; it is possible (Nagata 1959) to construct divisors on
surfaces which have positive intersection with every effective
divisor, but are not ample.)

Other criteria such as the Seshadri condition
give further characterisations of the ample cone.

## References

- Algebraic Geometry | year=1977}}
- On the 14th problem of Hilbert | year=1959 | journal=American Journal of Mathematics | issn=0002-9327 | volume=81 | pages=766–772}}

ample in Korean: 넉넉한 선다발

# Synonyms, Antonyms and Related Words

abounding, abundant, adequate, affluent, all-sufficing,
amplify, amplitudinous, aplenty, augment, barely sufficient,
bottomless, bounteous, bountiful, broad, capacious, commensurate, commodious, competent, complete, comprehensive, copious, corresponding, countless, decent, deep, detailed, develop, diffuse, dilate, distend, distended, due, effuse, elaborate, enlarge, enough, epidemic, equal to, exhaustless, expanded, expansive, extend, extended, extending, extensive, extravagant, exuberant, far-reaching,
fat, fertile, fit, flush, fruitful, full, galore, generous, good, good enough, great, handsome, in plenty, in
quantity, increase,
inexhaustible,
infinite, inflate, inflated, large, lavish, liberal, luxuriant, many, maximal, minimal, minimum, much, multitudinous, numerous, opulent, overflowing, plenitudinous, plenteous, plentiful, plenty, plenty good enough,
prevailing, prevalent, prodigal, productive, profuse, profusive, proportionable, proportionate, rampant, replete, rich, rife, riotous, roomy, running over, satisfactory, satisfying, spacious, spreading, substantial, sufficient, sufficient for,
sufficing, suitable, superabundant, swell, swollen, teeming, thorough, unfold, unsparing, unstinted, unstinting, up to, vast, voluminous, wealthy, well-found,
well-furnished, well-provided, well-stocked, wholesale, wide, wide-ranging, widespread