Nef line bundle

In algebraic geometry, a line bundle on a complete algebraic variety over a field is said to be nef if the degree of its restriction to every algebraic curve in the variety is non-negative. The term "nef" was introduced by Miles Reid^[1] 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". (A line bundle is called semi-ample or "eventually free" if some positive power is basepoint-free.) The older terminology was confusing because nef divisors are not the same as divisors numerically equivalent to effective divisors. For example, a curve with negative self-intersection number on a surface is effective but not nef.

Every semi-ample divisor is nef, but not every nef divisor is numerically equivalent to a semi-ample divisor, or even to an effective divisor. For example, Mumford constructed a line bundle L on a suitable ruled surface X such that L has positive degree on all curves, but the intersection number c₁(L)² is zero. It follows that L is nef, but no positive multiple of the first Chern class c₁(L) is numerically equivalent to an effective divisor.^[2] (The first Chern class is an isomorphism from the Picard group of line bundles on a variety X to the group of Cartier divisors modulo linear equivalence.)

A Cartier divisor D on an algebraic variety X is said to be nef if the corresponding line bundle O(D) is nef on X. Equivalently, D is nef if

$D\cdot C\geq 0\,$

for any algebraic curve C in X, in the sense of intersection theory.

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 N¹(X) of finite dimension, the Néron–Severi group tensored with the real numbers. The nef R-divisors form a closed convex cone in this vector space, called the nef cone. The interior of this cone is called the ample cone. For any projective variety X, Kleiman showed that a divisor is ample if and only if its numerical equivalence class lies in the interior of the nef cone.^[3] In particular, every ample line bundle is nef.

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 N₁(X) of 1-cycles modulo numerical equivalence. The vector spaces N¹(X) and N₁(X) are dual to each other by the intersection pairing, and the nef cone is the dual of the closure of the cone of curves. (The cone of curves need not be closed. For example, the class of the line bundle L on Mumford's surface is a 1-cycle which is not in the cone of curves, but is in its closure.)

Notes

1. M. Reid. Minimal models of canonical 3-folds. Algebraic Varieties and Analytic Varieties, 131-180. North-Holland (1983). Section 0.12f.
2. R. Lazarsfeld. Positivity in Algebraic Geometry I. Springer-Verlag (2004). Example 1.5.2.
3. R. Lazarsfeld. Positivity in Algebraic Geometry I. Springer-Verlag (2004). Theorem 1.4.23.

References

• Lazarsfeld, Robert (2004), Positivity in algebraic geometry, 1, Berlin: Springer-Verlag, ISBN 3-540-22533-1, MR 2095471
• Reid, Miles (1983), "Minimal models of canonical 3-folds", Algebraic Varieties and Analytic Varieties (Tokyo, 1981), Advanced Studies in Pure Mathematics, 1, North-Holland, pp. 131–180, ISBN 0-444-86612-4, MR 0715649
• Zariski, Oscar (1962), "The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface", Ann. of Math. (2), 76: 560–615, doi:10.2307/1970376, JSTOR 1970376, MR 0141668