Nef line bundle

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.Lazarsfeld (2004), Definition 1.4.1. (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" {{harv|Zariski|1962|loc=definition 7.6}} and "numerically effective", as well as for the phrase "numerically eventually free".Reid (1983), section 0.12f. 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.Lazarsfeld (2004), Example 1.4.5. More generally, a line bundle L is called semi-ample if some positive tensor power $L^\{\backslash otimes\; a\}$ 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 $D\backslash cdot\; C$ 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. Explicitly, the first Chern class $c\_1(L)$ is the divisor (s) of any nonzero rational section s of L.Lazarsfeld (2004), Example 1.1.5.

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 $N^1(X)$ of finite dimension, the Néron–Severi group tensored with the real numbers.Lazarsfeld (2004), Example 1.3.10. (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 $N^1(X)$, the nef cone Nef(X).

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 $N\_1(X)$ of 1-cycles modulo numerical equivalence. The vector spaces $N^1(X)$ and $N\_1(X)$ are dual to each other by the intersection pairing, and the nef cone is (by definition) the dual cone of the cone of curves.Lazarsfeld (2004), Definition 1.4.25.

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 $N^1(X)$ lies in the interior of the nef cone.Lazarsfeld (2004), Theorem 1.4.23. (An R-divisor is called ample if it can be written as a positive linear combination of ample Cartier divisors.) It follows from Kleiman's criterion that, for X projective, every nef R-divisor on X is a limit of ample R-divisors in $N^1(X)$. Indeed, for D nef and A ample, D + cA is ample for all real numbers c > 0.

Let X be a compact complex manifold with a fixed Hermitian metric, viewed as a positive (1,1)-form $\backslash omega$. Following Jean-Pierre Demailly, Thomas Peternell and Michael Schneider, a holomorphic line bundle L on X is said to be nef if for every $\backslash epsilon\; >\; 0$ there is a smooth Hermitian metric $h\_\backslash epsilon$ on L whose curvature satisfies

$\backslash Theta\_\{h\_\backslash epsilon\}(L)\backslash geq\; -\backslash epsilon\backslash omega$.

When X is projective over C, this is equivalent to the previous definition (that L has nonnegative degree on all curves in X).Demailly et al. (1994), section 1.

Even for X projective over C, a nef line bundle L need not have a Hermitian metric h with curvature $\backslash Theta\_h(L)\backslash geq\; 0$, which explains the more complicated definition just given.Demailly et al. (1994), Example 1.7.

• If X is a smooth projective surface and C is an (irreducible) curve in X with self-intersection number $C^2\backslash geq\; 0$, then C is nef on X, because any two distinct curves on a surface have nonnegative intersection number. If , then C is effective but not nef on X. For example, if X is the blow-up of a smooth projective surface Y at a point, then the exceptional curve E of the blow-up $\backslash pi\backslash colon\; X\backslash to\; Y$ has $E^2=-1$.
• Every effective divisor on a flag manifold or abelian variety is nef, using that these varieties have a transitive action of a connected algebraic group.Lazarsfeld (2004), Example 1.4.7.
• Every line bundle L of degree 0 on a smooth complex projective curve X is nef, but L is semi-ample if and only if L is torsion in the Picard group of X. For X of genus g at least 1, most line bundles of degree 0 are not torsion, using that the Jacobian of X is an abelian variety of dimension g.
• Every semi-ample line bundle is nef, but not every nef line bundle is even numerically equivalent to a semi-ample line bundle. For example, David 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 $c\_1(L)^2$ is zero.Lazarsfeld (2004), Example 1.5.2. It follows that L is nef, but no positive multiple of $c\_1(L)$ is numerically equivalent to an effective divisor. In particular, the space of global sections $H^0(X,L^\{\backslash otimes\; a\})$ is zero for all positive integers a.

A contraction of a normal projective variety X over a field k is a surjective morphism $f\backslash colon\; X\backslash to\; Y$ with Y a normal projective variety over k such that $f\_*\backslash mathcal\{O\}\_X=\backslash mathcal\{O\}\_Y$. (The latter condition implies that f has connected fibers, and it is equivalent to f having connected fibers if k has characteristic zero.Lazarsfeld (2004), Definition 2.1.11.) A contraction is called a fibration if dim(Y) < dim(X). A contraction with dim(Y) = dim(X) is automatically a birational morphism.Lazarsfeld (2004), Example 2.1.12. (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 $f^*(N^1(Y))\backslash subset\; N^1(X)$. Conversely, given the variety X, the face F of the nef cone determines the contraction $f\backslash colon\; X\backslash to\; Y$ up to isomorphism. Indeed, there is a semi-ample line bundle L on X whose class in $N^1(X)$ 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:Lazarsfeld (2004), Theorem 2.1.27.

:$Y=\backslash text\{Proj\; \}\backslash bigoplus\_\{a\backslash geq\; 0\}H^0(X,L^\{\backslash otimes\; a\}).$

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.Kollár & Mori (1998), Remark 1.26. (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.

Example: Let X be the blow-up of the complex projective plane $\backslash mathbb\{P\}^2$ at a point p. Let H be the pullback to X of a line on $\backslash mathbb\{P\}^2$, and let E be the exceptional curve of the blow-up $\backslash pi\backslash colon\; X\backslash to\backslash mathbb\{P\}^2$. Then X has Picard number 2, meaning that the real vector space $N^1(X)$ has dimension 2. By the geometry of convex cones of dimension 2, the nef cone must be spanned by two rays; explicitly, these are the rays spanned by H and H − E.Kollár & Mori (1998), Lemma 1.22 and Example 1.23(1). In this example, both rays correspond to contractions of X: H gives the birational morphism $X\backslash to\backslash mathbb\{P\}^2$, and H − E gives a fibration $X\backslash to\backslash mathbb\{P\}^1$ with fibers isomorphic to $\backslash mathbb\{P\}^1$ (corresponding to the lines in $\backslash mathbb\{P\}^2$ through the point p). Since the nef cone of X has no other nontrivial faces, these are the only nontrivial contractions of X; that would be harder to see without the relation to convex cones.

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Category:Geometry of divisors