MINIMAL MODELS IN NUMERICAL DIMENSION ONE
A plain-language edition for readers new to algebraic geometry
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(Original research paper rewritten to be clear, correct, concise,
approachable, elegant, and complete. The mathematics is unchanged;
only the presentation is made friendlier.)
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PART 0. READ THIS FIRST
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0.1 What is this paper about?
A lot of geometry is about shapes described by polynomial equations.
The sphere is the set of points (x,y,z) with x^2+y^2+z^2=1, and
x^2+y^2+z^2-1 is a polynomial. Shapes cut out by polynomial equations
are called algebraic varieties, and studying them is algebraic
geometry. That is the entire setting of this paper.
A complicated shape is often just a distorted version of a simpler one.
A dented beach ball and a perfect sphere are the same shape: you can
smooth the dents out without tearing anything. The minimal model
program (MMP) is a careful recipe for smoothing shapes out of a
variety while staying inside the world of polynomial varieties. Its
goal is a minimal model: the simplest variety in the same birational
family, one that cannot be smoothed any further by the allowed moves.
The big question this paper attacks:
If a variety bends the "right" way everywhere (technically: its
canonical divisor is pseudo-effective), must it have a minimal
model?
The expected answer is yes. This is the minimal model conjecture. It
is known for surfaces, for 3-folds, and for many higher-dimensional
families. This paper proves one more large case: when the bending has
a particular rank, called numerical dimension one. The main theorem
states this precisely in Part 1.
0.2 The key words, in plain language
variety: a shape defined by polynomial equations.
dimension: the number of independent directions you can move inside
it. A curve has dimension 1, a surface 2, a solid 3, and so on.
projective: the variety sits inside a projective space, the natural
compact home for these shapes. Think of a closed room with no exits,
so nothing runs off to infinity.
smooth: no sharp corners and no self-crossings.
divisor: a sub-shape of dimension one less (a curve on a surface),
or a formal sum of such sub-shapes with integer coefficients.
Think "a wall inside a room, counted with a multiplicity."
canonical divisor K_X: a special divisor attached to X that records
its curvature, or how it bends.
nef: short for numerically effective. A divisor D is nef if it meets
every curve in degree at least 0: D.C >= 0 for every curve C.
pseudo-effective: the divisor is a limit of genuinely positive ones.
It is slightly weaker than nef.
Q-factorial: every divisor can be multiplied by a positive integer to
become an honest divisor. Numerically it just lets us divide by
integers whenever needed.
terminal singularity: the mildest kind of dent. The MMP is allowed to
produce only terminal dents and nothing worse.
birational: two varieties are birational if they agree outside a small
bad set. You pass from one to the other by blowing up and blowing
down pieces. The MMP uses only birational changes.
0.3 The main theorem, in plain language
THEOREM (main result). Let X be a smooth connected complex projective
variety of dimension at least 3. Suppose K_X is pseudo-effective and
its numerical dimension kappa_sigma(X,K_X) equals 1. Then X has a
minimal model Y: a normal projective variety, Q-factorial, with only
terminal singularities, and with K_Y nef. It is reached from X by a
finite sequence of standard MMP moves. The map from X to Y contracts
no prime divisor of X, and the canonical divisors compare explicitly:
p*K_X = q*K_Y + E,
where p and q come from a common smooth model W down to X and Y, and
E is an effective exceptional divisor recording exactly which divisors
of X were contracted.
The hypothesis kappa_sigma = 1 is the technical heart. In words: when
you add one fixed ample twist to large multiples of K_X, the number of
independent sections grows no faster than linearly in the multiple.
Equivalently, kappa_sigma = 1 forbids QUADRATIC growth with a fixed
twist. The whole proof runs by contradiction: if a negative curve
existed, the sections WOULD grow quadratically, which the hypothesis
forbids. That is the strategy.
0.4 How the rest of the paper is organized
Part 1 (Introduction) states the theorem, defines kappa_sigma exactly,
states the good-minimal-model corollary, and surveys the literature.
Part 2 (Positive perturbations) builds one model Y carrying a whole
family of perturbed divisors K_Y + t*B_Y with t shrinking to zero.
Most of the MMP machinery lives here.
Part 3 (Two surface tests) is the numerical heart. One surface turns
positive square into section growth; a second surface forces a positive
square out of a hypothetical negative curve. Together they decide
nefness.
Part 4 (Conclusion) assembles the pieces to prove the main theorem.
Part 5 is a glossary. Turn back to it whenever a word is unfamiliar.
A note on difficulty. Parts 1 and 2, and all statements in Parts 3
and 4, use only high-school algebra. A few computations in Parts 3 and
4 (Nadel vanishing, the Hodge index theorem, Borel-Moore homology) are
standard graduate tools. Each is explained in words at the point of
use: what it says, why it applies, and what it delivers. You can follow
the logic without re-deriving those tools.
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PART 1. INTRODUCTION
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1.1 The minimal model problem
Fix a smooth projective variety X over the complex numbers. Two
positivity notions for a divisor D on X matter here.
- effective: D is a sum of irreducible divisors with nonnegative
coefficients, i.e. it is really "carved out" by some global
section rather than just a formal class.
- pseudo-effective: the numerical class of D lies in the closure of
the cone generated by effective divisors.
- nef: D has nonnegative degree on every integral curve C, that is
D.C >= 0 for all C.
The minimal model program replaces X by successive contractions and
flips along K_X-negative extremal rays. The models it produces are
normal (every local ring is integrally closed) and may carry terminal
singularities. A minimal model is required to:
(a) improve canonical discrepancies (its terminal singularities are
no worse than the ones X started with),
(b) extract no divisors (the inverse map contracts no prime divisor),
(c) have nef canonical divisor K_Y.
These requirements keep the birational information of the original
variety. The exact comparison of canonical divisors used here is part
of the theorem. We prove existence in the case where K_X has numerical
dimension one, measured by growth of sections with one fixed ample
twist.
1.2 The testing invariant kappa_sigma
For a Cartier divisor D on a smooth projective n-fold X, we use
Nakayama's section growth invariant:
kappa_sigma(X,D) = max { k in {0,...,n} :
there is an ample Cartier divisor A with
limsup_{m->oo} h^0(X, O(mD+A)) / m^k > 0 }.
The maximum is -infinity if the set is empty, and m runs over the
positive integers. The ample twist A is fixed BEFORE taking the limsup.
The key consequence: kappa_sigma(X,D) = 1 rules out positive quadratic
limsup for EVERY fixed ample Cartier twist A, including when the
quadratic growth is visible only along the multiples of one fixed
positive integer. It does NOT claim an upper bound of the shape
h^0(X, mD+A) = O(m).
This growth convention is distinct from the intersection formula for a
nef divisor,
nu(D) = max { k in {0,...,n} : D^k . H^{n-k} > 0 }, H ample.
We do not use that formula for K_X before we have built a nef model,
and we do not need a comparison theorem between the two notions of
numerical dimension.
1.3 The main theorem
THEOREM 1.1. Let X be a smooth connected complex projective variety of
dimension n >= 3. Suppose K_X is pseudo-effective and
kappa_sigma(X,K_X) = 1, with the convention of 1.2. Then there are a
normal projective Q-factorial terminal variety Y with K_Y nef and a
finite sequence
phi : X --> Y
of K-negative divisorial contractions and flips. The inverse of phi
contracts no prime divisor. On a common smooth projective resolution
p: W -> X and q: W -> Y, compatible canonical divisors satisfy
p*K_X = q*K_Y + E,
where E is an effective q-exceptional Q-divisor whose support contains
the strict transform of every prime divisor of X contracted by phi.
This settles the numerical-dimension-one case of the minimal-model
conjecture for smooth complex projective varieties of dimension at
least three. In particular it applies to smooth fivefolds, with no
hypothesis on the Euler characteristic chi(X, O_X).
1.4 Good minimal models
A good minimal model demands one more thing: a positive multiple of
K_Y generated by global sections. That is the semiampleness (or
abundance) conclusion. Nonvanishing asks only for a nonzero section of
some positive multiple of K. Theorem 1.1 asserts neither on its own;
the fixed ample twist from 1.2 is retained throughout the section
growth argument. A separate consequence uses log abundance.
COROLLARY 1.2 (good minimal model). Under the hypotheses of Theorem
1.1, the same Q-factorial terminal endpoint Y has semiample K_Y; hence
Y is a good minimal model of X.
Proof. Apply the log abundance theorem [OpenAILogAbundance2026,
Theorem 1.1] to the projective log-canonical (lc) pair (Y, 0) over C,
whose Q-Cartier canonical divisor is nef. QED
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1.5 The literature, and where the gap was
The three-dimensional program was built by Reid, Benveniste, Kawamata,
Shokurov, Kollar, Miyaoka and others. Mori's flip theorem [Mori88]
supplied a central step toward minimal-model existence in dimension 3;
see [KM98] for the broader development. The MMP replaces a variety by
successive contractions and flips along canonical-negative extremal
rays. The standard foundations, including the singularities and the
discrepancy comparisons used here, are in Kollar-Mori [KM98].
Birkar's formulation for log pairs [Birkar10, Conjecture 1.1] sets
existence of log minimal models beside the Mori fiber space
alternative. The decisive advance was Birkar, Cascini, Hacon and
McKernan [BCHM10]: minimal models exist for varieties of general type,
and klt pairs with big boundary admit finite MMP with scaling. Our
positive perturbations fall inside that established range. Their
boundary transforms stay big but need not stay ample.
At the opposite end, Nakayama developed the section-growth invariants
and divisorial Zariski decompositions behind numerical approaches to
the canonical divisor [Nak04]. Druel proved termination for directed
programs in numerical dimension zero [Druel11, Corollary 3.4], and
Gongyo extended the method to minimal-model existence for Q-factorial
dlt pairs of numerical log Kodaira dimension zero
[Gongyo11, Theorem 1.1].
The gap this paper closes: how to pass from arbitrarily small positive
perturbations to the canonical divisor itself in the next numerical
dimension. A finite program for each positive parameter does not, by
itself, terminate the concatenation of all of them at parameter zero.
The first reduction has a methodological predecessor. Alexeev, Hacon
and Kawamata use fourth homology to control certain four-dimensional
flips [AHK07, Lemma 3.1 and its proof]. We use the same kind of
decrease, after a discrepancy argument removes codimension-two
components on the flipped side. In dimension n, the relevant
finite-dimensional space is the span of algebraic (n-2)-cycle classes
in degree 2n-4. Lemma 2.3 below proves the needed statement directly;
the four-dimensional result is a precedent for the method, not an
imported termination theorem in all dimensions.
The second reduction combines two strands of positivity. Nadel's
multiplier-ideal vanishing theorem [Nadel90], in the formulation of
Lazarsfeld [Laz04II, Theorem 9.4.8] and [LazMI, Theorem 2.4], gives
the vanishing used to extend sections from a fixed very general
surface while letting the perturbation shrink with the section degree.
Surface Riemann-Roch then converts a positive square into quadratic
section growth. A different surface, drawn through a point of a
hypothetical negative curve, converts a vanishing-order estimate into
a fixed exceptional divisor. The Hodge index theorem makes its square
uniformly negative. The surface intersection calculations are
classical [Hartshorne77, Chapter V, Section 1]; keeping one surface
resolution and one exceptional curve fixed while the rational
perturbations vary is what produces the contradiction.
For comparison with results about sections of the unperturbed
canonical divisor: Lazic and Peternell prove nonvanishing for an
ALREADY MINIMAL projective terminal variety of numerical dimension one
with nonzero Euler characteristic [LP18, Theorem 6.7]. Liu and Xu prove
a good minimal model exists for smooth projective varieties of
dimension at most five and numerical dimension at most one, assuming
nonnegative Kodaira dimension [LiuXu25, Theorem 1.2]. They also obtain
a good minimal model for a smooth projective variety of dimension at
most four, with nonzero Euler characteristic and
0 <= kappa_sigma(X,K_X) <= 1 [LiuXu25, Corollary 5.2]. Those hypotheses
and conclusions differ from the existence statement here, and we do
not use their nonvanishing or abundance results as proof inputs.
1.6 The route through the proof
Step one: shrink perturbations on a single fixed model. Start with an
effective ample rational boundary B. Run finite minimal model programs
for decreasing positive coefficients of its transform. Every step is
K-negative. Discrepancy and cycle-class arguments show that only
finitely many steps change a locus of codimension at most two; they do
NOT assert termination of the concatenated programs. Proposition 2.1
then supplies a finite canonical prefix X --> Y such that multiples of
M_j = K_Y + t_j B_Y, t_j > 0, t_j -> 0,
are generated outside closed subsets Z_j of Y of codimension at least
three. Here B_Y is the transform of B; the generated multiple and the
bad set may depend on j.
Step two, two surfaces show K_Y is nef. Fix a very ample divisor H on
Y. A very general complete-intersection surface S avoids every Z_j and
the singular locus. The restrictions M_j|_S are semiample, so their
limit K_Y|_S is nef. Lemma 3.1 uses multiplier-ideal vanishing to
extend sections from this fixed surface with one fixed twist. A
positive square K_Y^2 . H^{n-2} would then force quadratic section
growth. Part 4 transfers that growth to X, where kappa_sigma(X,K_X)=1
rules it out. Hence the square is zero.
The first surface may miss a curve of negative K_Y-degree. To exclude
such a curve, even one inside the singular locus, take another
complete intersection from |H| through one of its points and pick a
surface component T through that point. On every component the
restriction of M_j has nonnegative square; these squares, counted with
multiplicity, sum to M_j^2 . H^{n-2} -> 0. So the square on T also
tends to zero. For all sufficiently small perturbations, an ambient jet
estimate on a resolution of Y forces sections of Cartier multiples
kM_j to vanish to order at least c k at one fixed point above the
curve, with c > 0 independent of j and k. On a fixed resolution of T
this yields an exceptional fixed part. The Hodge index theorem bounds
its square above by -epsilon k^2, with epsilon > 0 fixed, while the
residual moving system has nonnegative square. So the perturbations
have uniformly positive square on T, contradicting the zero-square
limit. This is the nefness criterion, Proposition 3.4. The finite
prefix already gives the remaining birational and canonical-comparison
claims of Theorem 1.1.
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1.7 Conventions
All varieties and morphisms are over C, and all birational models in
the proof are projective. We use compatible canonical divisors: on a
common resolution, the pullbacks of the canonical divisors on the two
models differ by a divisor supported on the exceptional locus.
Log discrepancies are normalized by
a(F; U, Delta) = 1 + coeff_F( K_{U~} - r*(K_U + Delta) ),
where r: U~ -> U is a resolution carrying the prime divisor F.
Terminal at zero boundary means a(F; U, 0) > 1 for every exceptional
prime F over U; an original prime divisor has log discrepancy 1.
For an effective rational boundary, klt ("Kawamata log terminal")
means all log discrepancies are positive [KM98, Chapter 2].
A birational map extracts no divisors if its inverse contracts no
prime divisor. Intersection numbers of rational Cartier divisors are
defined after clearing denominators. On an integral surface that need
not be normal, we use Cartier intersections with its fundamental cycle
[Fulton98, Chapter 2]. Characteristic-zero projective resolutions and
common resolutions are used throughout [Hironaka64].
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PART 2. POSITIVE PERTURBATIONS ON A FIXED MODEL
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The construction in this part does not use the numerical-dimension
hypothesis. It produces one model on which small positive
perturbations of the canonical divisor have base locus of codimension
at least three. That is what lets the very general surface S in Part 3
avoid every bad set at once.
2.1 The approximation proposition
PROPOSITION 2.1. Let X be a smooth projective variety of dimension
n >= 3 with pseudo-effective K_X. There are an effective ample
Q-divisor B on X, an integer s >= 0, and a finite sequence of
K-negative divisorial contractions and K-flips
X --> Y
with the following properties.
(i) Y is projective, Q-factorial and terminal. The map X --> Y
extracts no divisors.
(ii) Put L = K_Y, let B_Y be the transform of B, and set
t_j = 2^{-j}, M_j = L + t_j B_Y (j > s).
For each j > s there is a closed subset Z_j of Y of codimension
at least three such that |kM_j| is defined by a Cartier divisor
and is base point free on Y minus Z_j, for every sufficiently
divisible positive integer k.
(iii) On a common smooth projective resolution p: W -> X, q: W -> Y,
compatible canonical divisors satisfy
p*K_X = q*K_Y + E,
where E >= 0 is q-exceptional, and its support contains the
strict transform of every prime divisor of X contracted by
X --> Y.
Here and below, "sufficiently divisible" lets the required divisor
index depend on j. No uniform Cartier index for the infinite sequence
of models is needed.
The proof is 2.2-2.4. It has two ingredients: a discrepancy lemma that
records how singularities worsen along a flip, and a stabilization
lemma showing that after finitely many steps the moves no longer touch
codimension two.
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2.2 Discrepancies: how singularities change in one step
Write a(F,U) = a(F;U,0) with the normalization fixed in 1.7. The log
discrepancy measures how bad a singularity is: a larger value means a
milder singularity, and a smooth point scores the largest values of
all. The first lemma says a K-negative step never DECREASES a log
discrepancy, and that it strictly increases a discrepancy exactly on
the locus the step actually moves.
LEMMA 2.2 (Strict discrepancy increase). Suppose U --> U+ is a
K-negative divisorial contraction or a K-flip between normal projective
Q-Gorenstein varieties. Write h: U -> R for the contraction and
h+: U+ -> R for the other morphism (h+ is the identity in the
divisorial case). For every prime divisor F over these varieties,
a(F,U) <= a(F,U+).
The inequality is strict if the center of F on R lies in the locus
where either h or h+ has a positive-dimensional fiber.
Proof. Take a common smooth projective resolution W carrying F. Write
p: W -> U and q: W -> U+ for the projections, and g = h o p = h+ o q
for the common map to R:
W
/ \
p q
/ \
U - - - -> U+
\ /
h \ / h+
\ /
R
Set D = p*K_U - q*K_{U+}. This divisor is q-exceptional, because the
step extracts no divisors. On every curve contracted by g, the
divisors -p*K_U and q*K_{U+} have nonnegative degree; in the
divisorial case the second has degree zero. Hence -D is g-nef, and so
q-nef. The negativity lemma gives D >= 0; see [KM98, Lemmas
3.38-3.39].
Now the support assertion. Let r be a closed point of R with a
positive-dimensional fiber on either side. There is a curve in
g^{-1}(r) mapping onto a curve in that fiber: take a component
dominating that curve and cut it by sufficiently general ample
divisors. The relative ampleness signs give D.C < 0 for this lifted
curve. Consequently g^{-1}(r) meets Supp D.
We use the fiber-support form of the negativity lemma [KM98, Lemma
3.39]: the support of an effective divisor that is antinef over a
proper birational morphism to a normal variety either contains or is
disjoint from each fiber. In the present setting it can also be seen
directly on the reduced irreducible components of the fiber. If such
a component T is not contained in Supp D but meets it, a Cartier
multiple of D restricts to a nonzero effective Cartier divisor on the
integral projective variety T. Its degree against sufficiently many
ample hyperplanes is positive, producing a curve contracted by g with
positive D-degree. That contradicts antinefness. This argument does
not need T to be normal or the fiber to be reduced: only the support is
at issue. Components of dimension zero already lie in the support if
they meet it. The fibers of g are connected because R is normal and g
is proper birational [Hartshorne77, Corollary III.11.4]. Containment
therefore propagates through their reduced irreducible components.
It follows that every such fiber is contained in Supp D. If the center
of F on R lies in this locus, then F is contained in Supp D on our
chosen resolution. Finally,
a(F,U+) - a(F,U) = coeff_F D,
which proves both assertions. QED
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2.3 Stabilization: the steps stop touching codimension two
A terminal variety is already smooth in codimension two (Lemma 2.4
records this). The whole point of this part is to show that after
finitely many steps the MMP moves stop disturbing codimension two
entirely. Then a very general surface passes through every later stage
unchanged, which is exactly what Part 3 needs. Two mechanisms do the
work: a discrepancy count kills the codimension-two pieces of the
TARGET of each flip, and a homology count kills the codimension-two
pieces of the SOURCE.
LEMMA 2.3 (Stabilization in codimension two). Let
X = U_0 --> U_1 --> U_2 --> ...
be a finite or infinite sequence of K-negative divisorial contractions
and K-flips of projective Q-factorial varieties, starting from a
smooth n-fold, n >= 3. After finitely many steps, each remaining map
is an isomorphism outside closed subsets of codimension at least three
on both sides.
Proof. The usual MMP properties give preservation of
Q-factoriality, invariance of the Picard number under a flip, and a
drop of one under an elementary divisorial contraction [KM98,
Propositions 3.36-3.37]. There are therefore only finitely many
divisorial contractions. Since the steps extract no divisors, only
finitely many prime divisors of X can disappear; call this finite set
D.
All models are terminal. This follows from [KM98, Corollaries
3.42-3.43], or directly from Lemma 2.2: previously exceptional
valuations keep log discrepancy greater than one, and a newly
contracted divisor increases strictly from log discrepancy one.
Moreover, every valuation exceptional over the smooth X has integral
log discrepancy at least two. Discrepancies never decrease along the
sequence.
THE FLIPPED LOCI. Consider a flip whose target has a codimension-two
component T of its flipped locus. A terminal variety is smooth in
codimension two [KM98, Corollary 5.18]. Blowing up T at its generic
smooth point therefore defines a prime valuation F with
a(F,U_{i+1}) = 2.
Its center on the contraction base lies in the locus of
positive-dimensional fibers of the flipped contraction. Lemma 2.2
gives
a(F,X) <= a(F,U_i) < 2 = a(F,U_{i+1}). (*)
Thus F is not exceptional over X: it is an original prime divisor.
It is exceptional over U_{i+1}, so it belongs to D. After this
occurrence its discrepancy is at least two forever, and it cannot
satisfy (*) at any later step. So each flip with such a target
component uses a distinct member of the finite set D. After a finite
prefix, every remaining step is a flip and every flipped locus has
codimension at least three.
THE FLIPPING LOCI. We have removed codimension-two components on the
target side. We still must remove them on the source side; only then
can a general surface be carried unchanged through every later finite
stage. The argument adapts the homological decrease used in the proof
of [AHK07, Lemma 3.1], keeping only classes of algebraic cycles.
Put d = n - 2 and define the finite-dimensional real vector space
A_d(V) = span_R { [T] : T subset V is a closed integral d-fold }
subset H_{2d}(V,R).
Cycle classes, Borel-Moore localization, and their compatibility with
restriction to an open subset are as in [Fulton98, Section 19.1].
(The homology of compact complex algebraic varieties is
finite-dimensional and vanishes above their real dimension.)
For a remaining flip U --> U+ over R, remove the images in R of both
exceptional loci and take inverse images. This gives a common open
subset U0. Indeed, a proper birational morphism to a normal variety is
an isomorphism over its quasi-finite locus. Each exceptional locus
consists of positive-dimensional fibers; its image has dimension at
most one less than its own. Since the source contraction is small and
the target exceptional locus has dimension at most n-3, the closed
complements Z subset U and Z+ subset U+ satisfy
dim Z <= d, dim Z+ <= d - 1.
The extra points removed outside the exceptional loci lie in an
isomorphism locus and have dimension at most n-3.
Restriction gives maps
A_d(U) --> H^{BM}_{2d}(U0,R) <-- A_d(U+)
with the same image: intersect an integral d-fold with U0 and take its
closure on the other model, while a d-fold contained in the
complement restricts to zero. The right-hand map is injective, by the
localization sequence and H_{2d}(Z+,R)=0. Hence
dim A_d(U+) <= dim A_d(U). (**)
If the flipping locus has a d-dimensional component T, then [T] lies
in the kernel of the left-hand restriction, and it is nonzero: for an
ample Cartier divisor A,
<c_1(A)^d, [T]> = A^d . T > 0.
Then (**) is strict. A nonnegative integer can drop only finitely many
times, so eventually the flipping loci also have codimension at least
three. The same construction of the common open then gives this
codimension bound on both complements. QED
Lemma 2.4 will be used twice. We record it now because it is a
standard fact about terminal singularities.
LEMMA 2.4 (Terminal varieties are smooth in codimension two). A
terminal variety is smooth in codimension two [KM98, Corollary 5.18].
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2.4 Proof of Proposition 2.1, part 1: the finite stages
We build the model Y stage by stage, each stage a finite MMP with
scaling. First we choose a boundary that behaves well on every model.
CHOICE OF BOUNDARY. Choose an effective ample rational divisor B for
which (X,B) is klt and K_X + B is ample. For example, take a
sufficiently positive general smooth very ample divisor and multiply
it by a rational number strictly between zero and one. Since K_X is
pseudo-effective, K_X + tB is big for every rational t > 0.
SECTIONS PERSIST ON EVERY MODEL. We use a simple consequence of
normality. If a birational map X --> U extracts no divisors, then
every prime divisor of U corresponds to a prime divisor of X. For
compatible transformed divisors, the divisorial criterion for
regularity of a rational section gives an injection
H^0(X, kD) --> H^0(U, kD_U) (sec)
whenever the multiples are Cartier. So effectiveness and bigness of
B_U, and bigness of K_U + tB_U, persist on every model below. In
particular B_U need not be ample.
FINITE STAGES. Put t_j = 2^{-j} for every integer j >= 0. Starting
with V_0 = X, construct V_{j+1} from V_j by running the MMP for
K_{V_j} + Delta_j, Delta_j = t_{j+1} B_{V_j},
with scaling of
C_j = (t_j - t_{j+1}) B_{V_j}.
The precise input is [BCHM10, Corollary 1.4.2]: for a projective
Q-factorial klt pair (V, Delta) with big boundary Delta, and C >= 0
such that (V, Delta + C) is klt and K_V + Delta + C is nef, the MMP
with scaling of C terminates.
The inductive hypothesis is that (V_j, t_j B_{V_j}) is klt with nef
adjoint. The smaller pair is klt because B_{V_j} is effective; its
boundary is big by (sec); and the larger pair is exactly the one in
the inductive hypothesis. The adjoint stays big by (sec), so a Mori
fiber space cannot occur: an effective representative of a positive
multiple cannot have negative degree on curves covering general
fibers. Each stage is therefore a finite (possibly empty) sequence
ending at a nef adjoint. Standard MMP preservation gives the next klt
pair and a projective Q-factorial model. The rational divisor
K_{V_j} + t_j B_{V_j} is nef and big, hence semiample by the klt base
point free theorem [KM98, Theorem 3.3].
Now we must check that these adjoint steps really are canonical MMP
steps, so that Lemma 2.3 applies. That is the content of 2.5.
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2.5 Proof of Proposition 2.1, part 2: canonical steps and the fixed model
CANONICAL STEPS. To apply Lemma 2.3, every adjoint step must be a
canonical MMP step: canonical negativity on the contracted ray, and
(for a flip) canonical ampleness on the flipped side.
On its source U, let D = K_U + t_{j+1} B_U. The contracted ray R_0
satisfies
D . R_0 < 0, (D + lambda C) . R_0 = 0 (lambda > 0),
where C = (t_j - t_{j+1}) B_U is the current scaling divisor.
Therefore B_U . R_0 > 0 and K_U . R_0 < 0. This is the usual ray
calculation for a directed program [BCHM10, proof of Corollary 1.3.3],
and it already identifies each divisorial step as a K-negative
divisorial contraction.
For a small contraction h: U -> R, the adjoint flip makes D+
relatively ample. We show K_{U+} is relatively ample as well. Take
the positive rational number
b = (K_U . R_0) / (D . R_0).
The divisor K_U - bD is numerically trivial over R and descends up to
rational linear equivalence. To see this: a Cartier multiple is
relatively semiample by the relative klt base point free theorem for
the pair (U, t_{j+1} B_U) [KM98, Theorem 3.24]; subtracting its
adjoint D gives relative ampleness, because -D is relatively ample.
Its semiample morphism is constant on each connected projective
fiber, since its degree is zero on every fiber curve. The image is
proper and quasi-finite over R (each connected fiber maps to one
point), hence finite and birational over the normal R, hence equal to
R. Thus K_U - bD ~_Q h*A for a rational Cartier divisor A on R.
Both morphisms in the adjoint flip are small, so the relation
transforms to
K_{U+} - bD+ ~_Q (h+)*A.
Since D+ is relatively ample and b > 0, so is K_{U+}. The adjoint flip
is therefore a K-flip. All steps of the finite stages now satisfy the
hypotheses of Lemma 2.3.
THE FIXED MODEL. Concatenate the finite stages and apply Lemma 2.3.
Choose an endpoint Y = V_s after the resulting finite prefix. (If only
finitely many nonempty stages occur, choose s after the last one.) For
every j > s the finite composition Y --> V_j is an isomorphism
outside codimension-at-least-three closed subsets on both models. This
is preserved under finite composition: on a common open, remove the
next bad set and take its closure on the preceding model; its
dimension does not increase. Let Z_j be the resulting closed subset of
Y.
The divisor M_j = K_Y + t_j B_Y agrees on this common open with
K_{V_j} + t_j B_{V_j}. Choose a multiple that is Cartier on both
varieties and base point free on V_j. A line bundle on a normal
variety has the same global sections after deleting a subset of
codimension at least two, since it is reflexive. The two spaces of
sections agree, so the complete system on Y is generated outside Z_j.
This proves Proposition 2.1 (ii) for each j.
Finally, Y is projective, Q-factorial and terminal by construction,
and a finite composition of these steps extracts no divisors. On a
common smooth projective resolution of the finite chain, the effective
differences in Lemma 2.2 telescope to the comparison
p*K_X = q*K_Y + E. A divisor mapping onto a prime divisor of Y also
corresponds to a prime divisor of X, so its coefficient in E is zero.
Thus E is q-exceptional. For an original prime divisor contracted on
Y, terminality gives log discrepancy greater than one on Y, against
one on X; its strict transform has positive coefficient in E. This
proves Proposition 2.1 (iii). QED
A PLAIN-WORDS SUMMARY OF PART 2. We built one model Y on which the
divisors M_j = K_Y + t_j B_Y with t_j = 2^{-j} have their big
multiples generated by sections away from a codimension-three set Z_j.
The perturbations shrink to K_Y, and everything downstream only needs
those three properties plus the comparison of canonical divisors.
----------------------------------------------------------------
----------------------------------------------------------------
PART 3. TWO SURFACE TESTS FOR NEFNESS
----------------------------------------------------------------
This part is the numerical engine. It no longer uses the MMP. It only
takes one fixed model Y carrying the shrinking perturbations M_j from
Part 2, and decides whether L = K_Y is nef. The plan is two independent
tests, each carried out on a surface cut out of Y.
TEST 1 (Lemma 3.1). Take a very general surface S in Y. If the
number L^2 . H^{n-2} is positive, then the sections of L on S grow
like m^2. Back on the original X this quadratic growth contradicts
kappa_sigma(X,K_X)=1. So the number L^2 . H^{n-2} is zero.
TEST 2 (Proposition 3.4). The first surface is very general, so it
may simply miss a bad curve C_0 where L.C_0 < 0. Draw a second
surface T through one point of C_0. Sections that are negative along
C_0 must vanish at that point. That forced vanishing produces a
fixed sub-curve in T, and the Hodge index theorem then forces
positive square on T, again contradicting zero square.
Together: if L^2 . H^{n-2} = 0 and no negative curve exists, then L is
nef. That is the output of this part.
The two tests need precision about the perturbations. We use exactly
the following hypothesis, weaker than anything Part 2 actually
produced, so it is valid downstream.
THE APPROXIMATION HYPOTHESIS.
Y is a normal projective n-fold over C, n >= 3;
codim_Y (Sing Y) >= 3;
L and B_Y are Q-Cartier;
t_j are positive rationals with t_j -> 0;
M_j = L + t_j B_Y;
for each j there are a closed subset Z_j of Y with
codim_Y Z_j >= 3, and a positive integer d_j, such that
d_j M_j is Cartier and |d_j M_j| is generated by global
sections on Y minus Z_j.
Every positive multiple of d_j keeps the same generation property. The
integers d_j may depend on j. Below, restricting a rational Cartier
divisor means restricting it as a rational line bundle. Nothing here
assumes B_Y is effective or positive; the tests never need that.
We prove Lemmas 3.1-3.3 and Proposition 3.4 in turn.
----------------------------------------------------------------
3.1 A fixed twist detects positive surface square
The idea: a very general surface S misses every bad set. On S the
perturbations are honest semiample divisors, so the limit L|_S is nef
and L^2 . H^{n-2} >= 0. If that square is positive, surface
Riemann-Roch makes the section count grow like m^2, with one fixed
twist P that does not move as the perturbation shrinks.
LEMMA 3.1 (Surface growth with a fixed twist). Assume the
approximation hypothesis of Part 3, and let H be a very ample Cartier
divisor on Y. Fix a projective resolution pi: Yhat -> Y that is an
isomorphism over the smooth locus. There are a smooth
complete-intersection surface S subset Y, cut out by n-2 members of
|H|, and a Cartier divisor P on Yhat with the following properties.
Identify S with its inverse image in Yhat. Then L|_S is nef, so
L^2 . H^{n-2} >= 0. (nneg)
If d L is Cartier, then for every positive integer m divisible by d
the restriction map
H^0(Yhat, m pi*L + P) --> H^0(S, mL|_S + P|_S) (surj)
is surjective. In particular, as m -> oo through multiples of this one
fixed integer d,
h^0(Yhat, m pi*L + P) >= (m^2/2) L^2 . H^{n-2} + O(m), (quad)
where the error term depends only on the fixed surface and its
divisors.
Proof. Put r = n - 2. Very general members of |H| have a smooth
complete intersection S avoiding Sing Y and every Z_j. Indeed, each of
these countably many closed sets has dimension at most n-3, so the
condition that r hyperplanes meet any one of them is a proper closed
incidence condition. Over C these can be avoided simultaneously,
together with the closed conditions excluded by Bertini's theorem. The
inverse image of S is then exactly the smooth complete intersection cut
by the r pulled-back equations; there are no extra components lying
over the singular locus. Each M_j|_S is semiample, hence nef. Passing
to the limit on every curve of S shows L|_S is nef, proving (nneg).
Write A = pi*H, choose an ample Cartier divisor A_0 on Yhat, and make
the fixed choice
P = K_Yhat + rA + A_0. (twist)
Since A is nef and ampleness is open [Laz04I, Corollary 1.4.10 and
Theorem 1.4.23], there is eta > 0 such that
A_0 + (r - i)A - epsilon pi*B_Y (ample)
is ample whenever 0 <= i <= r and 0 <= epsilon < eta.
Now fix a positive m with d | m. Choose j = j(m) so that m t_j < eta,
then choose a multiple k of d_j with k > m. The pullback of |k M_j| is
generated near S. A general member D_m is smooth there by Bertini, and
R_m = (m/k) D_m >= 0, R_m ~_Q m pi*M_j, J(R_m) = O near S.
Here J(R_m) is the multiplier ideal; its local triviality follows from
m/k < 1 and the smoothness of D_m near S.
The divisors m pi*L + P - iA are integral Cartier divisors, and
m pi*L + P - iA - K_Yhat - R_m
~_Q A_0 + (r - i)A - m t_j pi*B_Y
is ample for 0 <= i <= r, by (ample). Nadel vanishing [Laz04II,
Theorem 9.4.8], equivalently [LazMI, Theorem 2.4], gives
H^q(Yhat, O(m pi*L + P - iA) (x) J(R_m)) = 0 (q > 0, 0 <= i <= r). (nadel)
Set F = O(m pi*L + P) (x) J(R_m). Tensoring the Koszul complex of the
r equations with F yields the exact complex
0 --> F(-rA) --> ... --> F(-A)^{oplus r} --> F
--> O_S(mL|_S + P|_S) --> 0.
At points of S the equations form a regular sequence and F is locally
free. Outside S one equation is a unit, so the Koszul complex is
contractible even after tensoring with an arbitrary sheaf. This proves
exactness everywhere. Splitting the complex into short exact sequences
and using (nadel) gives surjectivity on global sections onto its last
term. The inclusion F subset O(m pi*L + P) then proves (surj).
Adjunction and (twist) give P|_S - K_S = A_0|_S. Hence
m L|_S + P|_S - K_S is ample. Kodaira vanishing (the zero-boundary
ample case of [LazMI, Theorem 2.4]) and surface Riemann-Roch
[Hartshorne77, Chapter V, Section 1] give
h^0(S, m L|_S + P|_S)
= (m^2/2)(L|_S)^2 + (m/2) L|_S . (2P|_S - K_S)
+ (1/2) P|_S . (P|_S - K_S) + chi(S, O_S).
If S is disconnected, read the formula componentwise and sum. All
coefficients are fixed before m, j, k, D_m vary. Together with (surj),
this proves (quad). QED
----------------------------------------------------------------
3.2 A section cannot vanish too much along a curve
The second surface test reads a bad curve through the order to which
sections vanish at a point. Order of vanishing is measured in the local
ring of the ambient variety, so it survives even when the curve itself
is singular. The point of this lemma is a uniform bound: vanishing is
limited by how negative the line bundle is along the curve, and the
constant in that bound depends only on the curve and the variety, not
on the bundle.
LEMMA 3.2 (Uniform ambient multiplicity). Let V be a smooth projective
complex variety, let Gamma subset V be an integral curve, and let
iota: N -> V be the map from its normalization. For a section s, write
ord_z(s) for its order in the regular local ring of V at z. There is a
positive constant g, depending only on this map, such that for every
line bundle D on V, every nonzero section s in H^0(V, D), and every
closed point z in Gamma,
ord_z(s) >= - (deg_N iota*D) / g. (jet)
Proof. Choose a line bundle G of positive degree on the fixed smooth
curve N such that iota*(Omega^1_V) embeds as a subbundle of a direct
sum G^{oplus a} for some a. Such a choice follows by global generation
of (iota*(Omega^1_V))^* (x) G for a sufficiently ample G, then
dualization. Set g = deg G > 0.
For a nonzero s, the integer
b = min over closed p in Gamma of ord_p(s)
is finite and attained. The leading ambient jet gives a nonzero section
of
Sym^b(iota*(Omega^1_V)) (x) iota*D. (lead)
Here is the justification, valid even where Gamma is singular. For
b >= 1, the bundles of principal parts on the smooth variety V carry
the locally split exact sequence
0 -> Sym^b(Omega^1_V)(x)D -> P^b(D) -> P^{b-1}(D) -> 0.
The lower jet of s vanishes at every closed point of the reduced curve
Gamma. A section of a vector bundle on a reduced variety with this
property is zero, so that lower jet restricts to the zero section. The
principal-parts sequence is locally split as a sequence of O_V-modules,
so it stays exact after restriction to Gamma and pullback to N. The
restricted order-b jet therefore lies in the left-hand bundle. Its
value is nonzero at a point where the minimum b is attained; the
induced map on its fiber at any point of N above that point is an
isomorphism. Its pullback is thus regular and nonzero. For b = 0 use
s|_Gamma directly. This proves (lead) without assuming Gamma is smooth.
Taking symmetric powers of the subbundle inclusion above embeds (lead)
into a direct sum of copies of G^{tensor b} (x) iota*D. A nonzero
component is a nonzero section of this line bundle on N, so
0 <= b g + deg_N iota*D.
Since ord_z(s) >= b for every z in Gamma, (jet) follows. QED
----------------------------------------------------------------
3.3 A fixed exceptional curve forces positive square
This is the surface-side converse. If a fixed curve e on a resolution of
a surface T always receives a definite fraction of the fixed part of a
section system, then the surface square of the divisor cannot shrink to
zero. The fixed part is negative by the Hodge index theorem; the moving
part is nonnegative; their sum is the surface square, so that square
stays positive.
LEMMA 3.3 (Surface fixed-part estimate). Let T be an integral
projective surface, let f: T_1 -> T be a projective birational
morphism from a smooth surface, and fix a curve e subset T_1 contracted
by f. For each j, let D_j be a rational Cartier divisor on T, and let
k_j > 0 be an integer such that k_j D_j is Cartier. Suppose a nonzero
linear system of k_j f*D_j is generated outside the inverse image of a
finite subset of T. Let F_j be its fixed divisor. If, for one fixed
c > 0,
coeff_e F_j >= c k_j for all j, (fixcoef)
then there is epsilon > 0, independent of j, such that
D_j^2 . [T] >= epsilon for all j. (possq)
No normality assumption on T is needed.
Proof. Every curve in Supp F_j maps to a point of T, by the generation
assumption. Fix a very ample Cartier divisor H_T on T and an ample
Cartier divisor J on T_1, and put Q = f*H_T. The projection formula
gives
Q^2 = H_T^2 . [T] > 0, Q . F_j = f*D_j . F_j = 0. (orth)
The Hodge index theorem on the fixed smooth surface T_1 [Hartshorne77,
Chapter V, Section 1] makes the negative of the intersection form
positive definite on the orthogonal complement of Q in N^1(T_1, R).
Hence there is a single constant K > 0 such that
(J . F)^2 <= K (-F^2) for every F in that complement.
Effectiveness of F_j gives J . F_j >= c k_j (J . e). Therefore, with
the fixed positive number epsilon = c^2 (J . e)^2 / K,
F_j^2 <= - epsilon k_j^2. (negsq)
After removing F_j, the residual linear system has no fixed curve. Its
square is nonnegative: choose two members with no common curve and take
their effective intersection, or use a nowhere-vanishing member if the
system is trivial. By (orth) and the projection formula,
0 <= (k_j f*D_j - F_j)^2 = k_j^2 (D_j^2 . [T]) + F_j^2.
Now (negsq) proves (possq). QED
----------------------------------------------------------------
3.4 The nefness criterion
Now we combine everything. This is the keystone: zero surface square
plus the two lemmas force L nef. The proof is a contradiction. A
hypothetical negative curve C_0 forces a second surface T to have a
uniformly positive square, which the zero-square hypothesis forbids.
PROPOSITION 3.4 (Zero square implies nef). Assume the approximation
hypothesis of Part 3. If, for one very ample Cartier divisor H on Y,
L^2 . H^{n-2} = 0, (zerosq)
then L is nef.
Proof. Suppose an integral curve C_0 subset Y has L . C_0 < 0. Fix a
projective resolution pi: Yhat -> Y that is an isomorphism over the
smooth locus. Choose a closed point x in C_0 outside the images of
those irreducible components of pi^{-1}(C_0) that do not dominate C_0.
There are only finitely many such images, and each is a point.
Consequently every point of pi^{-1}(x) lies on a component of
pi^{-1}(C_0) that dominates C_0.
A SURFACE THROUGH x. Choose n-2 very general members H_1, ..., H_{n-2}
of |H|, all passing through x. They meet properly, and their
intersection meets each Z_j and Sing Y in at most finitely many points.
(The hyperplanes through x have no other base point; successive
general cuts avoid all positive-dimensional pieces of the relevant
intersections. The countably many conditions can again be imposed
simultaneously over C.) Write their intersection cycle as
H_1 ... H_{n-2} . [Y] = sum_i a_i [T^(i)], a_i > 0,
with each T^(i) an integral surface. The dimension theorem guarantees a
component through x; fix one and call it T.
On every T^(i), the system induced by |d_j M_j| has at most a finite
base locus. Its square is nonnegative, including when T^(i) is
nonnormal. To check this directly: choose a section nonzero at the
generic point, then a second section not vanishing identically on any
curve of the first zero divisor. The resulting Cartier intersections
with [T^(i)] form an effective zero-cycle. If the first section has no
zero, the restricted line bundle is trivial and its square is zero.
These are the usual Cartier intersection and projection formulas on
cycles [Fulton98, Chapter 2]. It follows that M_j^2 . [T^(i)] >= 0 and
sum_i a_i (M_j^2 . [T^(i)]) = M_j^2 . H^{n-2} --> 0
by (zerosq). In particular
M_j^2 . [T] --> 0. (Tlim)
FIXED GEOMETRY ABOVE x. The surface T meets Sing Y only in finitely
many points. Its strict transform T_Yhat is birational to T and maps
onto it. Choose any z in T_Yhat above x. There is an integral curve
Gamma subset Yhat through z dominating C_0. Indeed, take an irreducible
component A subset pi^{-1}(C_0) through z. By the choice of x, A
dominates C_0. If dim A = d, its fiber A_x is a proper closed subset
of dimension at most d-1. General very ample cuts through z, in number
d-1, cut A_x down to dimension at most zero and leave a curve component
through z on A. That component cannot lie in A_x, so it dominates C_0.
For d = 1, take Gamma = A.
Let iota: N -> Yhat denote the normalization of Gamma followed by
inclusion. Resolve T_Yhat by a smooth projective surface T_0 ->
T_Yhat, choose a point w in T_0 above z, and blow it up. Write T_1 for
the resulting surface, e subset T_1 for the exceptional curve of this
blowup, and f: T_1 -> T for the composite map. The surface T_1, the
curve e, and f are fixed throughout the rest of the argument.
THE CONTRADICTION ON THE FIXED SURFACE. The projection formula and
M_j -> L give, for one fixed delta > 0,
deg_N iota* pi* M_j <= - delta (negdeg)
for all sufficiently large j. (This degree is the positive degree of
N -> C_0 times M_j . C_0.) For each such j, choose any positive
multiple k_j of d_j. Lemma 3.2, applied on the fixed Yhat and Gamma,
shows that every nonzero pullback section s' in H^0(Yhat,
k_j pi* M_j) of a section on Y satisfies
ord_z(s') >= c k_j, c = delta / g > 0. (vanord)
The constant c depends only on the fixed normalized curve and the
negative-degree margin in (negdeg).
The sections on Y induce a nonzero linear system of k_j f*(M_j|_T),
generated outside f^{-1}(T intersect Z_j). Nonzeroness follows from
generation at points of T minus Z_j. Every nonzero section in this
induced system has order at least c k_j at w before the last blowup:
the local ring map O_{Yhat,z} -> O_{T_0,w} sends the maximal ideal
power into the matching power; apply (vanord) to a section inducing it.
Sections whose restriction is identically zero do not contribute to the
induced system. Hence its fixed divisor F_j satisfies
coeff_e F_j >= c k_j.
All hypotheses of Lemma 3.3 hold with D_j = M_j|_T. Lemma 3.3 gives a
fixed epsilon > 0 with M_j^2 . [T] >= epsilon for all sufficiently
large j, contradicting (Tlim). Therefore no curve C_0 of negative
L-degree exists, and L is nef. QED
----------------------------------------------------------------
----------------------------------------------------------------
PART 4. CONCLUSION: PROVING THE MAIN THEOREM
----------------------------------------------------------------
We now apply the two halves. Part 2 gives the fixed model Y with
shrinking perturbations; Lemma 3.1 turns positive surface square into
quadratic section growth; the growth carries back to X, where
kappa_sigma(X,K_X)=1 forbids it. That kills the square, and Proposition
3.4 then makes K_Y nef. The birational and canonical-comparison claims
were already built into Y.
PROOF OF THEOREM 1.1. Apply Proposition 2.1 to obtain a finite canonical
MMP
X --> Y
and the divisors L = K_Y, B_Y, and M_j = L + t_j B_Y, with generated
multiples outside closed subsets Z_j of codimension at least three. The
variety Y is terminal, hence smooth in codimension two (Lemma 2.4).
Those are exactly the standing hypotheses of the numerical results in
Part 3.
Fix a very ample Cartier divisor H on Y and a projective resolution
pi: Yhat -> Y that is an isomorphism over the smooth locus. Lemma 3.1
gives a fixed Cartier divisor P on Yhat. Choose a positive integer r_0
with r_0 L Cartier. Then
h^0(Yhat, m pi*L + P) >= (m^2/2) L^2 . H^{n-2} + O(m)
for r_0 | m, (quadratic)
and also L^2 . H^{n-2} >= 0.
Suppose this intersection number is positive. Choose a smooth
projective common resolution W dominating Yhat and all the models in
the finite MMP prefix, with maps
r: W -> Yhat, u: W -> X.
Proposition 2.1 gives
u*K_X - r* pi* L >= 0.
For every positive multiple m of r_0, this effective difference gives
an injection
H^0(Yhat, m pi*L + P) --> H^0(W, m u*K_X + r*P).
Set P_X = u_* r* P. This is a fixed integral Weil divisor, hence
Cartier on the smooth X. A rational section f in the right-hand space
satisfies
div_W(f) + m u*K_X + r*P >= 0.
Pushing this inequality forward gives
div_X(f) + m K_X + P_X >= 0.
So there is an injective linear map
H^0(W, m u*K_X + r*P) --> H^0(X, m K_X + P_X).
Choose a single ample Cartier divisor A_X for which A_X - P_X has a
nonzero section. Multiplication by that fixed section gives
h^0(X, m K_X + A_X) >= h^0(Yhat, m pi*L + P).
By (quadratic), the left-hand side is bounded below by c m^2, for a
fixed c > 0, along all sufficiently large multiples of r_0. Hence
limsup_{m -> oo} h^0(X, m K_X + A_X) / m^2 > 0,
contradicting kappa_sigma(X,K_X) = 1 with the convention of 1.2. It
follows that L^2 . H^{n-2} = 0.
Proposition 3.4 now implies L = K_Y is nef, including on curves lying
in the singular locus. The remaining conclusions of Theorem 1.1,
including the effective exceptional comparison p*K_X = q*K_Y + E and
its strict support condition, are supplied by the finite MMP in
Proposition 2.1. QED
This completes the proof of the main theorem. Corollary 1.2 then adds
semiampleness of K_Y by log abundance, upgrading the minimal model to a
good minimal model.
----------------------------------------------------------------
----------------------------------------------------------------
PART 5. GLOSSARY AND REFERENCES
----------------------------------------------------------------
5.1 Glossary
abundance: the assertion that a nef canonical divisor is semiample
(a positive multiple is generated by global sections).
ample: a divisor whose high multiples separate points and give an
embedding into projective space. Stronger than nef.
base locus / base point free: the common zero set of all sections of a
linear system; base point free means that set is empty.
big: a divisor whose sections grow as fast as possible, like m^n on an
n-fold.
birational: two varieties that agree outside a small bad set, related
by blowing up and blowing down.
Borel-Moore homology: a homology for possibly noncompact spaces, used
for the localization sequence in the codimension-two count.
canonical divisor K_X: the divisor of the top-degree differential form;
it encodes curvature. For a smooth variety over C, c_1(T*X) = -K_X.
Cartier divisor: a divisor given locally by one equation, so it can be
pulled back and intersected. Q-Cartier means a positive multiple is
Cartier.
complete intersection: the common zero set of the right number of
general equations; here a surface cut by n-2 members of |H|.
discrepancy, log discrepancy: a number measuring how singular a pair is.
Smooth is best. Terminal and klt are the two mild classes used here.
divisorial contraction: an MMP step that contracts a divisor.
effective: a genuine nonnegative sum of prime divisors.
exceptional locus: the set a birational morphism collapses.
extract no divisors: the inverse map contracts no prime divisor.
flip: an MMP step replacing a small contraction by another small
contraction, resolving a K-negative curve with no divisor to contract.
good minimal model: a minimal model whose canonical divisor is
semiample.
Hodge index theorem: on a surface, the intersection form has exactly one
positive direction (up to sign conventions); used to control squares
of fixed curves.
klt (Kawamata log terminal), lc (log canonical): standard singularity
classes for log pairs, defined by the signs of log discrepancies.
kappa_sigma, numerical dimension: growth-based measures of positivity,
defined by counting sections of large multiples with one fixed ample
twist.
minimal model: a birational model with nef canonical divisor that
extracts no divisors.
MMP, minimal model program: the recipe of successive K-negative
contractions and flips.
multiplier ideal: an ideal measuring the singularities of an effective
Q-divisor; Nadel vanishing is its vanishing theorem.
nef: nonnegative degree on every curve.
normalization: the resolution of a curve or variety by a smooth/normal
one, one-to-one away from the singular set.
pseudo-effective: a limit of effective divisors; weaker than nef.
semiample: a positive multiple is globally generated.
terminal: the mildest singularity class produced by the MMP (smooth in
codimension two, log discrepancies greater than one).
5.2 References
AHK07 V. Alexeev, C. Hacon, Y. Kawamata. Termination of (many)
4-dimensional log flips. Invent. Math. 168 (2007), 433-448.
BCHM10 C. Birkar, P. Cascini, C. Hacon, J. McKernan. Existence of
minimal models for varieties of log general type. J. Amer. Math.
Soc. 23 (2010), 405-468.
Birkar10 C. Birkar. On existence of log minimal models. Compositio
Math. 146 (2010), 919-928.
Druel11 S. Druel. Quelques remarques sur la decomposition de Zariski
divisorielle. Math. Z. 267 (2011), 413-423.
Fulton98 W. Fulton. Intersection Theory, 2nd ed. Springer, 1998.
Gongyo11 Y. Gongyo. On the minimal model theory for dlt pairs of
numerical log Kodaira dimension zero. Math. Res. Lett. 18
(2011), 991-1000.
Hartshorne77 R. Hartshorne. Algebraic Geometry. Springer, 1977.
Hironaka64 H. Hironaka. Resolution of singularities of an algebraic
variety over a field of characteristic zero. Ann. Math. 79
(1964), 109-326.
KM98 J. Kollar, S. Mori. Birational Geometry of Algebraic Varieties.
Cambridge Tracts 134, 1998.
Laz04I R. Lazarsfeld. Positivity in Algebraic Geometry I. Springer,
2004.
Laz04II R. Lazarsfeld. Positivity in Algebraic Geometry II. Springer,
2004.
LazMI R. Lazarsfeld. A short course on multiplier ideals. IAS/Park
City, 2010.
LiuXu25 J. Liu, Z. Xu. Non-vanishing implies numerical dimension one
abundance. arXiv:2505.05250, 2025.
LP18 V. Lazic, T. Peternell. Abundance for varieties with many
differential forms. Epijournal Geom. Algebrique 2 (2018), art. 1.
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