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).
Roughly speaking, positivity properties of a line bundle are related to having many global sections.
In more detail, a line bundle is called basepoint-free if it has enough sections to give a morphism to projective space.
The notions described in this article are related to this construction in the case of a morphism to projective space with
the line bundle on projective space whose global sections are the homogeneous polynomials of degree 1 (that is, linear functions) in variables
The degree of a line bundle L on a proper curve C over k is defined as the degree of the divisor (s) of any nonzero rational section s of L. The coefficients of this divisor are positive at points where s vanishes and negative where s has a pole.
As a result, a basepoint-free line bundle L on any proper scheme X over a field is nef, meaning that L has nonnegative degree on every (irreducible) curve in X.
[6] Analogously, in complex geometry, Cartan's theorem A says that every coherent sheaf on a Stein manifold is globally generated.
A line bundle L on a proper scheme over a field is semi-ample if there is a positive integer r such that the tensor power
[8] The latter definition is used to define very ampleness for a line bundle on a proper scheme over any commutative ring.
[11] Ample line bundles are used most often on proper schemes, but they can be defined in much wider generality.
The basic characterization of ample invertible sheaves states that if X is a quasi-compact quasi-separated scheme and
The rest of this article will concentrate on ampleness on proper schemes over a field, as this is the most important case.
(For example, if X is smooth over k, then a Cartier divisor can be identified with a finite linear combination of closed codimension-1 subvarieties of X with integer coefficients.)
Here s may depend on F.[17][18] Another characterization of ampleness, known as the Cartan–Serre–Grothendieck theorem, is in terms of coherent sheaf cohomology.
Namely, a line bundle L on a proper scheme X over a field (or more generally over a Noetherian ring) is ample if and only if for every coherent sheaf F on X, there is an integer s such that for all
[19][18] In particular, high powers of an ample line bundle kill cohomology in positive degrees.
To determine whether a given line bundle on a projective variety X is ample, the following numerical criteria (in terms of intersection numbers) are often the most useful.
means the associated Cartier divisor (defined up to linear equivalence), the divisor of any nonzero rational section of L. On a smooth projective curve X over an algebraically closed field k, a line bundle L is very ample if and only if
[25] The Nakai–Moishezon criterion (named for Yoshikazu Nakai (1963) and Boris Moishezon (1964)) states that a line bundle L on a proper scheme X over a field is ample if and only if
Kleiman's criterion states that a line bundle L on X is ample if and only if L has positive degree on every nonzero element C of the closure of the cone of curves NE(X) in
Equivalently, a line bundle is ample if and only if its class in the dual vector space
[28] A line bundle on a projective variety is called strictly nef if it has positive degree on every curve.
Nagata (1959) and David Mumford constructed line bundles on smooth projective surfaces that are strictly nef but not ample.
C. S. Seshadri showed that a line bundle L on a proper scheme over an algebraically closed field is ample if and only if there is a positive real number ε such that deg(L|C) ≥ εm(C) for all (irreducible) curves C in X, where m(C) is the maximum of the multiplicities at the points of C.[30] Several characterizations of ampleness hold more generally for line bundles on a proper algebraic space over a field k. In particular, the Nakai-Moishezon criterion is valid in that generality.
[33] On a projective scheme X over a field, Kleiman's criterion implies that ampleness is an open condition on the class of an R-divisor (an R-linear combination of Cartier divisors) in
In terms of divisors with integer coefficients (or line bundles), this means that nH + E is ample for all sufficiently large positive integers n. Ampleness is also an open condition in a quite different sense, when the variety or line bundle is varied in an algebraic family.
For example, a vector bundle F is ample if and only if high symmetric powers of F kill the cohomology
[39] A useful weakening of ampleness, notably in birational geometry, is the notion of a big line bundle.
This is the maximum possible growth rate for the spaces of sections of powers of L, in the sense that for every line bundle L on X there is a positive number b with