In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, 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
In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every .
...
, 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
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals ...
-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 fo ...
scheme A scheme is a systematic plan for the implementation of a certain idea.
Scheme or schemer may refer to:
Arts and entertainment
* ''The Scheme'' (TV series), a BBC Scotland documentary series
* The Scheme (band), an English pop band
* ''The Schem ...
''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.
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 Pe ...
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
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
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 Mumf ...
''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 t ...
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
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 Mumfo ...
. Explicitly, the first Chern class
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 of finite dimension, the
Néron–Severi group 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
, 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
of 1-cycles modulo numerical equivalence. The vector spaces
and
are
dual
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 (grammatical ...
to each other by the intersection pairing, and the nef cone is (by definition) the
dual cone of the cone of curves.
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
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
. 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 with a fixed
Hermitian metric, viewed as a positive
(1,1)-form . Following
Jean-Pierre Demailly
Jean-Pierre Demailly (25 September 1957 – 17 March 2022) was a French mathematician who worked in complex geometry. He was a professor at Université Grenoble Alpes and a permanent member of the French Academy of Sciences.
Early life and educ ...
, Thomas Peternell and Michael Schneider, a
holomorphic line bundle ''L'' on ''X'' is said to be nef if for every
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 algebraic ...
Hermitian metric
on ''L'' whose
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canon ...
satisfies
.
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
, 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
, 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
''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 Hemmings ...
of a smooth projective surface ''Y'' at a point, then the exceptional curve ''E'' of the blow-up
has
.
*Every effective divisor on a
flag manifold or
abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functio ...
is nef, using that these varieties have a
transitive action 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.
...
.
*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 ''g'' at least 1, most line bundles of degree 0 are not torsion, using that the
Jacobian
In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to:
* Jacobian matrix and determinant
* Jacobian elliptic functions
* Jacobian variety
*Intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähle ...
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
is zero. It follows that ''L'' is nef, but no positive multiple of
is numerically equivalent to an effective divisor. In particular, the space of global sections
is zero for all positive integers ''a''.
Contractions and the nef cone
A
contraction
Contraction may refer to:
Linguistics
* Contraction (grammar), a shortened word
* Poetic contraction, omission of letters for poetic reasons
* Elision, omission of sounds
** Syncope (phonology), omission of sounds in a word
* Synalepha, merged ...
of a
normal projective variety ''X'' over a field ''k'' is a surjective morphism
with ''Y'' a normal projective variety over ''k'' such that
. (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: ...
. Conversely, given the variety ''X'', the face ''F'' of the nef cone determines the contraction
up to isomorphism. Indeed, there is a semi-ample line bundle ''L'' on ''X'' whose class in
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:
:
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
at a point ''p''. Let ''H'' be the pullback to ''X'' of a line on
, and let ''E'' be the exceptional curve of the blow-up
. Then ''X'' has Picard number 2, meaning that the real vector space
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
, and ''H'' − ''E'' gives a fibration
with fibers isomorphic to
(corresponding to the lines in
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