More generally, a line bundle L on a proper scheme X over a field k is said to be nef if it has nonnegative degree on every (closed irreducible) curve in X.
[1] (The degree of a line bundle L on a proper curve C over k is the degree of the divisor (s) of any nonzero rational section s of L.) A line bundle may also be called an invertible sheaf.
The term "nef" was introduced by Miles Reid as a replacement for the older terms "arithmetically effective" (Zariski 1962, definition 7.6) and "numerically effective", as well as for the phrase "numerically eventually free".
Every line bundle L on a proper curve C over k which has a global section that is not identically zero has nonnegative degree.
As a result, a basepoint-free line bundle on a proper scheme X over k has nonnegative degree on every curve in X; that is, it is nef.
[3] More generally, a line bundle L is called semi-ample if some positive tensor power
To go back from line bundles to divisors, the first Chern class is the isomorphism from the Picard group of line bundles on a variety X to the group of Cartier divisors modulo linear equivalence.
is the divisor (s) of any nonzero rational section s of L.[4] To work with inequalities, it is convenient to consider R-divisors, meaning finite linear combinations of Cartier divisors with real coefficients.
The R-divisors modulo numerical equivalence form a real vector space
of finite dimension, the Néron–Severi group tensored with the real numbers.
[5] (Explicitly: two R-divisors are said to be numerically equivalent if they have the same intersection number with all curves in X.)
An R-divisor is called nef if it has nonnegative degree on every curve.
The nef R-divisors form a closed convex cone in
The cone of curves is defined to be the convex cone of linear combinations of curves with nonnegative real coefficients in the real vector space
[6] A significant problem in algebraic geometry is to analyze which line bundles are ample, since that amounts to describing the different ways a variety can be embedded into projective space.
One answer is Kleiman's criterion (1966): for a projective scheme X over a field, a line bundle (or R-divisor) is ample if and only if its class in
Let X be a compact complex manifold with a fixed Hermitian metric, viewed as a positive (1,1)-form
Following Jean-Pierre Demailly, Thomas Peternell and Michael Schneider, a holomorphic line bundle L on X is said to be nef if for every
When X is projective over C, this is equivalent to the previous definition (that L has nonnegative degree on all curves in X).
[8] Even for X projective over C, a nef line bundle L need not have a Hermitian metric h with curvature
[9] A contraction of a normal projective variety X over a field k is a surjective morphism
[13] (For example, X could be the blow-up of a smooth projective surface Y at a point.)
A face F of a convex cone N means a convex subcone such that any two points of N whose sum is in F must themselves be in F. A contraction of X determines a face F of the nef cone of X, namely the intersection of Nef(X) with the pullback
Conversely, given the variety X, the face F of the nef cone determines the contraction
is in the interior of F (for example, take L to be the pullback to X of any ample line bundle on Y).
Any such line bundle determines Y by the Proj construction:[14] To describe Y in geometric terms: a curve C in X maps to a point in Y if and only if L has degree zero on C. As a result, there is a one-to-one correspondence between the contractions of X and some of the faces of the nef cone of X.
[15] (This correspondence can also be formulated dually, in terms of faces of the cone of curves.)
Knowing which nef line bundles are semi-ample would determine which faces correspond to contractions.
The cone theorem describes a significant class of faces that do correspond to contractions, and the abundance conjecture would give more.
Then X has Picard number 2, meaning that the real vector space