Numerically effective
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In algebraic geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
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 In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...
(built from codimension-1 subvarieties), there is an equivalent notion of a nef divisor.


Definition

More generally, a line bundle ''L'' on a
proper Proper may refer to: Mathematics * Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact * Proper morphism, in algebraic geometry, an analogue of a proper map for ...
scheme ''X'' over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''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 In mathematics, an invertible sheaf is a coherent sheaf ''S'' on a ringed space ''X'', for which there is an inverse ''T'' with respect to tensor product of ''O'X''-modules. It is the equivalent in algebraic geometry of the topological notion of ...
. The term "nef" was introduced by
Miles Reid Miles Anthony Reid FRS (born 30 January 1948) is a mathematician who works in algebraic geometry. Education Reid studied the Cambridge Mathematical Tripos at Trinity College, Cambridge and obtained his Ph.D. in 1973 under the supervision of P ...
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 In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
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 L^ 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 In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mu ...
''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 In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for ta ...
D\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 In mathematics, the Picard group of a ringed space ''X'', denoted by Pic(''X''), is the group of isomorphism classes of invertible sheaves (or line bundles) on ''X'', with the group operation being tensor product. This construction is a global ve ...
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''.


The nef cone

To work with inequalities, it is convenient to consider R-divisors, meaning finite linear combinations of Cartier divisors with
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
coefficients. The R-divisors modulo numerical equivalence form a real
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
N^1(X) of finite dimension, the
Néron–Severi group In algebraic geometry, the Néron–Severi group of a variety is the group of divisors modulo algebraic equivalence; in other words it is the group of components of the Picard scheme of a variety. Its rank is called the Picard number. It is nam ...
tensored with the real numbers. (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 In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety X is a combinatorial invariant of importance to the birational geometry of X. Definition Let X be a proper variety. By definition, a (real) ''1-cycle'' ...
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 Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatica ...
of the cone of curves. A significant problem in algebraic geometry is to analyze which line bundles are
ample 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). The most important notion of positivity is that of a ...
, 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. (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.


Metric definition of nef line bundles

Let ''X'' be a
compact complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a c ...
with a fixed
Hermitian metric In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on ea ...
, viewed as a positive (1,1)-form \omega. Following Jean-Pierre Demailly, Thomas Peternell and Michael Schneider, a
holomorphic line bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...
''L'' on ''X'' is said to be nef if for every \epsilon > 0 there is a
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebrai ...
Hermitian metric h_\epsilon on ''L'' whose curvature satisfies \Theta_(L)\geq -\epsilon\omega. When ''X'' is projective over C, this is equivalent to the previous definition (that ''L'' has nonnegative degree on all curves in ''X''). Even for ''X'' projective over C, a nef line bundle ''L'' need not have a Hermitian metric ''h'' with curvature \Theta_h(L)\geq 0, which explains the more complicated definition just given.


Examples

*If ''X'' is a smooth projective surface and ''C'' is an (irreducible) curve in ''X'' with self-intersection number C^2\geq 0, then ''C'' is nef on ''X'', because any two ''distinct'' curves on a surface have nonnegative intersection number. If C^2<0, then ''C'' is effective but not nef on ''X''. For example, if ''X'' is the
blow-up ''Blowup'' (sometimes styled as ''Blow-up'' or ''Blow Up'') is a 1966 mystery drama thriller film directed by Michelangelo Antonioni and produced by Carlo Ponti. It was Antonioni's first entirely English-language film, and stars David Hemming ...
of a smooth projective surface ''Y'' at a point, then the exceptional curve ''E'' of the blow-up \pi\colon X\to Y has E^2=-1. *Every effective divisor on a
flag manifold In mathematics, a generalized flag variety (or simply flag variety) is a homogeneous space whose points are flags in a finite-dimensional vector space ''V'' over a field F. When F is the real or complex numbers, a generalized flag variety is a smo ...
or abelian variety is nef, using that these varieties have a
transitive action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphis ...
of a connected
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...
. *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 Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...
in the Picard group of ''X''. For ''X'' of
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
''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 David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded t ...
constructed a line bundle ''L'' on a suitable
ruled surface In geometry, a surface is ruled (also called a scroll) if through every point of there is a straight line that lies on . Examples include the plane, the lateral surface of a cylinder or cone, a conical surface with elliptical directrix, t ...
''X'' such that ''L'' has positive degree on all curves, but the intersection number c_1(L)^2 is zero. 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^) is zero for all positive integers ''a''.


Contractions and the nef cone

A contraction of a
normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...
projective variety ''X'' over a field ''k'' is a surjective morphism f\colon X\to Y with ''Y'' a normal projective variety over ''k'' such that f_*O_X=O_Y. (The latter condition implies that ''f'' has
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
fibers, and it is equivalent to ''f'' having connected fibers if ''k'' has characteristic zero.) A contraction is called a fibration if dim(''Y'') < dim(''X''). A contraction with dim(''Y'') = dim(''X'') is automatically a birational morphism. (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 In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
f^*(N^1(Y))\subset N^1(X). Conversely, given the variety ''X'', the face ''F'' of the nef cone determines the contraction f\colon X\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 In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not funct ...
: :Y=\text\bigoplus_H^0(X,L^). 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''. (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 In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety X is a combinatorial invariant of importance to the birational geometry of X. Definition Let X be a proper variety. By definition, a (real) ''1-cycle'' ...
describes a significant class of faces that do correspond to contractions, and the
abundance conjecture In algebraic geometry, the abundance conjecture is a conjecture in birational geometry, more precisely in the minimal model program, stating that for every projective variety X with Kawamata log terminal singularities over a field k if the canonic ...
would give more. Example: Let ''X'' be the blow-up of the complex projective plane \mathbb^2 at a point ''p''. Let ''H'' be the pullback to ''X'' of a line on \mathbb^2, and let ''E'' be the exceptional curve of the blow-up \pi\colon X\to\mathbb^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\to\mathbb^2, and ''H'' − ''E'' gives a fibration X\to\mathbb^1 with fibers isomorphic to \mathbb^1 (corresponding to the lines in \mathbb^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.


Notes


References

* * * * *{{Citation , authorlink=Oscar Zariski , mr=0141668 , last=Zariski , first=Oscar , title=The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface , journal=Annals of Mathematics , series=2 , volume=76 , year=1962 , pages=560–615 , doi=10.2307/1970376 Geometry of divisors