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Concept
Nef line bundle
Summary
In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line bundles are described by a convex cone, and the possible contractions of the variety correspond to certain faces of the nef cone. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of a nef divisor.
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. (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" and "numerically effective", as well as for the phrase "numerically eventually free". The older terms were misleading, in view of the examples below.
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. More generally, a line bundle L is called semi-ample if some positive tensor power is basepoint-free. It follows that a semi-ample line bundle is nef. Semi-ample line bundles can be considered the main geometric source of nef line bundles, although the two concepts are not equivalent; see the examples below.
A Cartier divisor D on a proper scheme X over a field is said to be nef if the associated line bundle O(D) is nef on X. Equivalently, D is nef if the intersection number is nonnegative for every curve C in X.
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.
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