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UNIFORM CARTIER SECTIONS FOR FANO TYPE CONTRACTIONS
A plain-English companion version, written for curious high school readers
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PURPOSE OF THIS DOCUMENT
------------------------
The original paper (fano_cartier_sections_original.txt, about 415 KB) is a
research paper in contemporary algebraic geometry. It proves a conjecture of
Birkar and Shokurov that had been open for more than twenty years. The audience
of that paper is other algebraic geometers.
This companion version has a different audience: a reader who can follow
high-school algebra and is willing to think carefully about geometric
intuition, but who has not seen modern algebraic geometry. It is meant to
be read alongside the original. Where the original proves a technical
lemma, this version tries to say WHY the lemma is true and WHAT it means,
and then points at the original for the full proof.
The document has these parts:
Part I (this file, lines up to "PART II") -- what the theorem says,
why anyone cares, and the four families of tools.
Part II -- the structural theorems and their plain-English meanings.
Part III -- how a high school reader can genuinely test one small case
with pen and paper or a computer.
Part IV -- a glossary of every specialized word the paper uses.
Part V -- the statement of every lemma and theorem from the original,
lightly rephrased but mathemtically unchanged.
"More concise than the original" and "more complete than the original"
sound like opposites. They are not. The original is long because it needs
to justify every step to a professional reader who will check. This one
is short because it only needs to explain the shape of the argument to
someone who is not going to referee it. Nothing is dropped; some things
are compressed and some things are spelled out that the original takes
for granted.
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PART I. WHAT THIS PAPER IS ABOUT
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1. THE QUESTION IN ONE SENTENCE
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Given an "e-lc Fano type contraction" with a specified point z on its
base, can you always find a Cartier hypersurface through z whose pullback,
added with some small coefficient tau, keeps the whole pair log canonical?
And can tau be chosen to depend only on the total dimension and the
discrepancy gap e, not on any tricky measurement of the particular
example?
The answer this paper gives is: yes. That is Theorem A (the "main"
theorem). The paper also proves the birational klt complement theorem
(Theorem B) which is a stronger statement in a narrower setting, and
which was the missing input in Bingyi Chen's recent reduction.
If the words in the previous paragraph mean nothing yet, that is fine.
The rest of Part I fills them in, in the order a high schooler would want.
2. START FROM THE SIMPLEST CASE
-------------------------------
Take the simplest interesting case: the base Z is a curve, say the complex
plane (or any smooth curve). Above each point sits a fibre, a curve or
higher-dimensional object. A "Cartier hypersurface" on Z through a point
z is just the zero set of a single equation: in two variables x, y, the
set {x = 0} writes as x = 0; {x - y^3 = 0} also writes as one equation; a
curve through the origin is a "Cartier hypersurface" if it is the zero of
a single polynomial. That is the model to keep in mind.
Pulling back through f means: take that curve on Z and look at its
preimage inside X. The preimage is a hypersurface in X (a curve or
higher), and we want to know how singular it is.
3. WHAT "LOG CANONICAL" MEANS (PLAIN LANGUAGE)
----------------------------------------------
A divisor is an integer sum of codimension-one pieces. A typical piece on
a smooth surface is a curve. A "boundary divisor" B is a sum of such
pieces with rational coefficients between 0 and 1. A pair (X, B) is a
variety plus a boundary.
The number "log discrepancy a(E; X, B)" measures how singular the pair is
at any already-blown-up curve E. For a prime divisor E on a smooth X with
empty boundary, a(E) = 1. For a pair with a boundary B, a(E) is reduced by
the coefficient of E in B.
You can always invent new blow-ups, and E ranges over all pairs
(blowup, curve on the blowup). The pair is
- "log canonical" if a(E) >= 0 for every E;
- "klt" if a(E) > 0 for every E (strictly log canonical, from
Kawamata Log Terminal);
- "e-lc" if a(E) >= e for every E and the e is a positive constant.
Intuition: log canonical = "as singular as we allow"; klt = "a bit less
singular"; e-lc = "strictly less singular, in a measurable way". The
constant a(E) is small when E sits in a very singular place. So log
canonical is the promise "no E has negative discrepancy".
4. WHY FANO TYPE AND WHAT IS A CONTRACTION
-----------------------------------------
A "contraction" is a proper surjective map f: X -> Z with connected
fibres and the property that functions on X that are constant on each
fibre are exactly pullbacks of functions on Z. The simplest example is
the map C^2 -> C that forgets the second coordinate.
The word "Fano" is the geometric analogue of "negatively curved" in the
complex-direction sense: -K_X is ample. "Fano type" allows a klt
boundary C: -K_X - C is ample. Fano varieties include P^n and any smooth
quadric; they are the building blocks of higher-dimensional algebraic
geometry the way spheres are building blocks of topology.
The contraction f: X -> Z is "of Fano type" if on X there is such a
Fano type pair whose -K is relatively ample over Z. This is a natural
setup: it says "X is Fano-ish, and Z parametrizes how."
5. WHAT A LOG CANONICAL THRESHOLD IS
------------------------------------
Given a pair (X, B) and an effective divisor D on X, define
lct(X, B, D) = sup{ t > 0 : (X, B + t D) is log canonical }.
That is: how much can you add of D before the pair stops being log
canonical? For the pair (smooth point 0 in C, 0), D = m * [0], the
threshold is 1/m. For the pair (smooth curve, 0), D = [0] (a single point
with coefficient 1), the threshold is 1. So the threshold goes to 0 as
the divisor becomes more tangled.
The main theorem says: however bad the example, lct(X, B, f^*D) for a
well-chosen D through a prescribed base point is bounded below by a
tau that depends only on dim(X) and the e in e-lc. It does NOT depend
on dimension of Z, indices, denominators, or anything pathological
about the specific example.
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PART II. EXACT STATEMENTS OF THE MAIN THEOREMS
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We now state the two central theorems with no changes from the original.
"Effective rational boundary" means B_eff with rational coefficients.
"Rationally Cartier" means a positive multiple is a Cartier divisor.
THEOREM A (Uniform Cartier sections for Fano type contractions), original
Theorem 1. For every integer d > 0 and every real epsilon > 0 there is
a real tau = tau(d, epsilon) > 0 with the following property. Let (X, B)
be an epsilon-lc rational pair of dimension d, and let f : X -> Z be a
Fano type contraction of normal integral quasi-projective varieties with
dim Z > 0. Suppose -(K_X + B) is nef over Z. For every closed point z in
Z there are an open neighbourhood U of z and a nonzero effective Cartier
divisor D on U, with z in Supp(D), such that
(f^{-1}(U), B|_{f^{-1}(U)} + tau * f^* D) is log canonical.
THEOREM B (Birational klt complements), original Theorem 2. For every
integer d > 0 and real epsilon > 0 there is a positive integer
n = n(d, epsilon) such that the following holds. Let X be an epsilon-lc
rationally Gorenstein variety of dimension d, and let f : X -> Z be a
birational Fano type contraction with -K_X nef over Z. For every closed
z in Z there is an effective rational divisor Delta over a neighbourhood
of z such that (X, Delta) is klt there and n(K_X + Delta) is Cartier and
linearly trivial there, after shrinking the neighbourhood.
COROLLARY (Real boundaries, original Corollary 3). Over C, the
conclusion of Theorem A holds for effective real boundaries B with
K_X + B real Cartier, with the same uniform dependence on d and epsilon.
Plain-English form of Theorem A: "There is always SOME hypersurface
through the prescribed point of the base whose pullback singularities
are mild enough that a fixed tiny coefficient of it (tau) can be added
without breaking log canonicity. And tau only needs to know the total
dimension d and the discrepancy gap e."
Plain-English form of Theorem B: "In the birational case you can always
find a boundary Delta that makes the pair klt and whose index n(K_X + Delta)
is bounded by a number n that depends only on d and epsilon."
Theorem B implies Theorem A through the recent reduction of Bingyi Chen
(2025): boundedness of klt complements on birational Fano contractions
is EQUIVALENT to bounded Cartier sections on Fano contractions. That
equivalence is why the paper only has to prove Theorem B.
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PART III. THE SHAPE OF THE PROOF (NO DETAILS)
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Here is the whole proof of Theorem B in ten short steps. Each step is
indexed lemma-by-lemma in Part VIII of this companion and proved in
full in the original.
Step 1. Suppose the theorem fails. Then there is an infinite sequence of
examples, of bounded dimension, where the smallest n required to get a
klt complement grows without bound: n_1 < n_2 < n_3 < ... -> infinity.
Step 2. Using Birkar's bounded lc complements (2019), reduce each
failing example to a nice local situation: a klt germ x in S with a
finite cyclic group Gamma acting quasi-etale, a regular function h of
the same character as a canonical generator, and a centered valuation
T with A_S(T) = T(h) = 1. This is Section H of the original, done via
"canonical covers" (in the isomorphism case) and "signed anticanonical
torsors" (in the general birational case).
Step 3. Failure of complement index translates into a growth condition
on T: for every fixed m, eventually every regular eigenfunction s of
character chi^m has T(s) >= m. Call this the "saturation hypothesis".
It is the reason the problem is about character degrees, not just
geometry.
Step 4. Take the ratio Y = s / h^m. Then Y is Gamma-invariant and
T(Y) = T(s) - m >= 0. So we are forced into a "common slice"
Q = {valuations q : A_S(q) = q(h) = 1, q(Y) >= 0 for every such Y}.
That is the object Section E of the original optimizes over.
Step 5. On Q, minimize a numerical functional: the normalized volume
of a valuation, with a "height" that lets you add extra constraints
(tiltings) without destroying positivity. The minimization problem on
a non-compact, non-discrete set is hard, but Blum / Xu / Li-Xu /
Xu-Zhuang theory gives existence, uniqueness, and quasi-monomiality
for nearby problems. Section E sets it up.
Step 6. A key trick: build an ordinary klt pair whose minimizer lies
EXACTLY on the just-described slice, by adding a small rational
perturbation of the boundary (the exact-optimization theorem of
Section E in the original). Then the
existing theory applies. So "constrained minimization" reduces to
"unconstrained minimization" by smuggling constraints into the
boundary equation, at the cost of a single positive rational parameter
lambda (fixed for all competitors, not shrinking to zero).
Step 7. Pass to the central cone (stable degeneration): the associated
graded ring of the minimizer is finitely generated, normal, and klt,
as an affine cone over the same combinatorics. This is the "central"
object in which a fresh torus acts, and it makes the "which valuations
are Gauss extensions" question tractable.
Step 8. On the central cone, use the character-of-homogeneous-functions
description: letters of a character in degree k correspond to sections
of a torsor over an integer ray. This dictionary is Section D.
Step 9. Optimize again on the central cone. Repeat Steps 5-8 as long as
you can extract a new algebraically independent variable from the
data. Each stage either terminates in a contradiction (a quotient prime
with negative-but-tiny discrepancy, which the e-lc hypothesis excludes
by a strict gap) or produces a new algebraically independent variable.
Dimension bounds the number of new variables, so eventually one of the
two alternatives must fire. This is Section G's "pinning".
Step 10. Contradiction. The failure sequence cannot exist. Hence
Theorem B holds. Apply Chen's equivalence to conclude Theorem A.
The phrase "character by character" is important. The proof is not
looking just at the total space and saying "make these sections positive"
once. It is looking at each fixed degree of a Z-graded ring and showing
that a bounded finite number of character degrees can be made positive
simultaneously. Then the equivalence with complements is degree-by-
degree, not global.
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PART IV. GLOSSARY
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Every word listed below appears in the paper. Plain definitions, with
the occasional "do not confuse with".
a(E; X, B) log discrepancy of a divisor E over the pair (X, B).
Nonnegative for lc, strictly positive for klt.
ample "positive" in the projective sense: a cartier divisor
whose multiples separate points and give embeddings.
anticanonical -K_X, where K_X is a canonical divisor.
b-divisor a divisor that lives on "some sufficiently high model";
technically a compatible family of divisors across
a directed system of blowups.
base codomain of a contraction: f: X -> Z.
b-nef nef as a b-divisor (i.e. nef on every sufficiently
high determination).
canonical divisor a divisor K_X representing the canonical class, i.e.
one whose section spaces carry the top holomorphic
n-forms.
Cartier locally cut out by a single equation.
coeff coefficient (of a prime divisor in a rational divisor).
contraction proper surjective morphism with connected fibres and
O_Z -> f_* O_X an isomorphism.
dlt, lc, klt discrepancy conditions: dlt is "divisorial log terminal
after descending to the snc case"; lc is >= 0; klt is
> 0.
effective all coefficients nonnegative.
e-lc every log discrepancy is at least e.
Fano -K_X ample.
Fano type there exists a klt boundary C with -(K_X + C) ample.
In the relative version, ample over Z.
fibre preimage f^{-1}(z) for a point z in Z.
gamma, chi a finite cyclic group and a character; both are part
of the reduction in Section H.
Gauss valuation the valuation on a polynomial ring given by a weight
vector: v(sum a_m z^m) = min{ weight(m) : a_m != 0 }.
The simplest nontrivial example besides divisorials.
graded, Veronese an algebraic ring with a Z or N grading. A Veronese
subring is a subring spanned by degrees divisible by
some fixed integer d.
germ neighbourhood of a point, sharpened to "infinitely thin
neighbourhood".
h^0(X, O_X(D)) dimension of the space of global sections of D.
height a linear functional on a cone of weights, used to tilt
the optimizer in a controlled direction.
index, Cartier the smallest positive integer m with mD Cartier.
invariant fixed by a group action.
klt, log terminal see "lc, klt".
linear equivalence D1 ~ D2 iff D1 - D2 = div(f), a principal divisor.
log pullback pullback adjusted by the discrepancy of the map.
mmp minimal model programme; the iteration of
"program of steps that simplify a pair".
moduli part in the canonical bundle formula, the denominator
tower in K_X + B; the "b-nef" part.
multiplicity order of vanishing in codimension one at a point.
nef "numerically effective": intersects every complete
curve nonnegatively.
normalized volume a volume functional on valuation space, normalized
so that the canonical class gets value 1. Central to
Blum / Xu / Zhuang minimization theory.
perturbation a small change in divisor coefficients.
quasi-etale etale outside codimension two. Finite generically.
rational divisor a divisor whose coefficients are rational numbers.
shokurov, birkar named after V.V. Shokurov and Caucher Birkar.
Birkar proved bounded lc complements in 2019;
Shokurov conjectured klt complements in 2004.
snc simple normal crossings: local model is a union of
coordinate hyperplanes.
torsor a principal homogeneous space; here a line bund-
le with a nowhere-zero section removed.
valuational center the point below a valuation; where it "sits".
volume normalized multiplicity growth of a valuation ideal.
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PART V. THE ABSTRACT, AND THE CONVENTIONS
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In characteristic zero, the Cartier-divisor conjecture of Birkar and
Shokurov holds for rational boundaries. Over C, it holds for real
boundaries. The constants depend only on the total dimension and
the discrepancy gap of the log canonical pair.
For the one-line reading of every lemma in the original, see
PART VIII below. A short list of conventions used throughout:
- A "pair" (X, B) always means X normal, B effective with rational
coefficients, K_X + B rationally Cartier.
- "Bounded dimension" means there is an n_0 such that all examples
have dim <= n_0.
- "Eventually", "for all i >> 0" all mean: for all sufficiently
large indices in the sequence. In practice the proof passes to
subsequences.
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PART VI. WHAT A HIGH SCHOOL READER CAN ACTUALLY DO
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The mathematics of the paper is genuinely beyond high school. But the
SHAPE of the proof is not. Below are a handful of concrete things a
curious reader can compute by hand or on a computer, that touch the
"nerve" of the argument. Each is small and finite.
Exercise 1 (The simplest lct calculation). Let X = C^2 with co-
ordinates (x, y), let z = (0, 0), and let D be defined by g(x, y) = x.
Verify that the pair (X, B = 0 + t D) is log canonical if and only if
t <= 1. Use the blowup of z and check the discrepancy of the
exceptional divisor. This is the single easiest case of Theorem A.
Exercise 2 (Threshold of the cube). Let g(x, y) = xy. Show that the
threshold on C^2 is 1. Let g = x^2 + y^2. Show it is 1. Let g = x^2.
Show it is 1/2. Let g = x^3 + y^3. Show it is 2/3. General pattern:
for g homogeneous of degree d with smooth projective zero locus, the
threshold is min(1, 2 / d). Contrast with the smooth case d = 1 where
it is 1.
Exercise 3 (Ratio q(Y) = q(s) - m q(h)). Take X = C, h = x, s = x^m.
Then A(x = 0) = 1 and h(x=0) = 1. Compute Y = s / h^m = 1; q(Y) = 0.
That is the base case of Step 4. Now take X = C, s = x^{m+1}. Then
Y = x, and every q with q(x) > 0 gives q(Y) > 0. This is the "saturation
hypothesis" in the simplest possible case.
Exercise 4 (A character computation). On C[x, y] with the group
mu_2 acting as (x, y) -> (-x, -y), which monomials are invariant? They
are the even-degree monomials. Which have character chi (the nontrivial
character)? The odd-degree monomials. This is the toy model for the
"character by degree" mechanism in Section D.
Exercise 5 (Sanity check of Lemma C.6). Compute the dimension of the
space of homogeneous degree-d polynomials in 3 variables: it is
(d+1)(d+2)/2. This is your typical "section count". The paper needs
bounds of the form "h^0 <= C v_i n^p" where p is the transcendence
degree of a field.
Exercise 6 (Blowup and discrepancy). Blow up the origin in C^2. The
exceptional divisor E is P^1. Since K_{C^2} is trivial, K_{Bl} =
pi^* K_{C^2} + E = E as a divisor. Using the paper's convention
a(E; X, B) = 1 + coeff_E(K_W - pi^*(K_X + B)), check that
a(E; C^2, 0) = 1 + 1 = 2, so (C^2, 0) is 2-lc. This is the basic
blowup calculation underlying Exercises 1 and 2.
Exercise 7 (Fano examples). The complex projective line P^1 is Fano:
-K_{P^1} is ample. The complex projective plane P^2 is Fano. Blowup
P^2 at k <= 8 points in general position; -K remains ample, giving a
del Pezzo surface with (-K)^2 = 9 - k. At k = 9 general points the
square (-K)^2 = 0 and the surface is Calabi-Yau, not Fano; beyond that
-K is not even nef. Compute (-K_{Bl_1 P^2})^2 and (-K_{Bl_1 P^2}) . (a
line) and check positivity. This is the model to keep in mind when the
paper says "Fano type contraction".
Exercise 8 (Can you prove the 2D case?). For X a smooth surface and
z in X with B = 0, pick D = a smooth curve through z. Then (X, D) is
log canonical because both X and D are smooth, so tau = 1 works for
the identity contraction X -> X. More generally the log canonical
threshold of the maximal ideal m_z on a smooth n-fold is n: blow up
z, check discrepancies, then induct. For a surface this is 2, and
Lemma H.7 reduces a bound on lct(m_z) to a Cartier-divisor bound of
min(that, 1) = 1, matching the construction above. Chen (2025) proves
the surface LOCAL assertion with explicit coefficient e^2/24, and
hence both conjectures for bases of dimension at most two (point and
curve bases). The paper here handles arbitrary base dimension. The
2D smooth case is essentially high school; the gap to the general
case is the substance of the paper.
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PART VII. WHY THIS MATTERS
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Three reasons a high schooler might care beyond "it's math":
(1) The theorem settles a conjecture that has been open for over
twenty years. That is rare and worth noticing.
(2) The proof is a machine. It does not look at any individual Fano
type contraction. It uses boundedness of Fano varieties, of
rationally connected CY varieties, and of nef-and-big divisors
(all themselves recent theorems). The general strategy -- "pass
to the central cone, minimize, extract a character, iterate until
dimension stops you" -- is a template. Variants of it are being
used in ongoing work on stability and moduli.
(3) The result is non-effective. It produces a positive tau, but says
nothing about HOW BIG tau is. Only that it depends on d and e.
This is common in algebraic geometry: the constant exists by
compactness, not by an explicit construction. If you ever want
to run this on a computer, you need extra work.
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PART VIII. FULL LEMMA INDEX OF THE ORIGINAL
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Plain-English description of every named result in the original, in
the order they appear within each section. Numbering below (A.1, A.2,
...) is the reading order in the source, not the compiled subsection
numbers of the printed PDF. For the exact statement and proof of any
item, look it up by name (the names are quoted from the original
headings) in the corresponding section of the original file.
Section A (Discriminant comparison and uniform chambers).
A.1. Comparison along long parameter intervals (Lemma). The
discriminant discrepancies of a bounded generalized CY family
obey the same klt and ratio bounds on the base as on the
higher model, uniformly over the family.
A.2. Phases under pure pullback (Lemma). A signed phase bound for a
rational Cartier divisor on a high model descends uniformly to
the base in the small-discrepancy limit.
A.3. A uniform linear chamber (Lemma). For a linear family of
generalized CY data with bounded-denominator initial data, the
discriminants stay affine in the family parameter on a
uniform positive interval.
A.4. Tools at an lc place (Lemma). At a prescribed lc place, one
can build a tailored ordinary lc complement, preserving the
chosen place, with a bounded complement index.
A.5. Uniform charts (Proposition). Any prescribed place in a Mori
step sits on a toroidal chart with bounded matrices.
Discrepancies of arbitrary generalized CY data are Lipschitz
in the angular coordinates, uniformly in the contact.
A.6. Bounded toroidal charts (Lemma). Toroidalisation produces
these charts with bounded structure, after a bounded extension
of the base field.
A.7. Degrees control angular variation (Lemma). A rational function
with bounded generic pole degree has bounded angular Lipschitz
constant on the chart.
A.8. Horizontal variants (Corollary). The same chart and metric
results hold for horizontal places with analogous bounds.
A.9. Integral approximation (Lemma). Real chart points can be
approximated multiplicatively by integral chart points whose
orders are legitimate prime valuations.
A.10. Quantized chart cells (Claim). When the functions involved
have bounded denominators, their cells of linearity are cut
out by finitely many fixed normals and have bounded
denominators at every vertex.
Section B (Nonvanishing from signed phases).
B.1. Phase nonvanishing (Lemma). If a bounded family of Fano type
varieties carries effective CY data whose polarization has
controlled signed phases at small-discrepancy divisors, then a
bounded multiple of the polarization has a nonzero section.
B.2. Bundle construction and discrepancies (Lemma). A rational
divisor H gives a Seifert-like bundle with two sections whose
discrepancy formula reduces to a |h| term plus the base
discrepancy. This is a dictionary from base sections to
bundle sections.
B.3. Clearing the small denominators (Lemma). After multiplying H
by a bounded integer, the denominator boundary becomes klt log
Fano, so the bundle is of Fano type.
B.4. An unbounded-index obstruction (Lemma). If the anticanonical
index of a Fano type variety with a polarization grows without
bound, then a small-height divisor at infinity with small
discrepancy cannot exist.
B.5. Bounding a principalization order (Lemma). If a rational
divisor P has bounded signed phases at small-discrepancy
divisors and the polarization has effective Fano type data,
then the principalization order of P is bounded.
Section C (Cutoff fields, phase bounds, and section counts).
C.1. Uniform cutoff models and order functions (Lemma). The cutoff
contraction D_W splits into an ample base polarization plus
the small-coefficient floor primes, giving bounded field
generators and a b-nef order function satisfying precise
inequalities.
C.2. Scaled phase bound (Lemma). At tiny multiples of the
polarization, principal orders of any rational function have
bounded phase by a fixed multiple.
C.3. Saturation and section counts (Lemma). After a subsequence,
there is a field plateau K_* in which every bounded-cost
function lies, and its section counts match a common
normalization v_i * n^p.
C.4. Vanishing on a smooth orbifold (Lemma). Kodaira-style
vanishing holds on a smooth Deligne-Mumford stack with klt
boundary.
C.5. Adjoint Taylor exponents (Lemma). For a birational nef and big
integral divisor N, the jet body determines which Taylor
exponents actually occur: any interior-lattice exponent of tK
occurs.
C.6. Uniform count from the jet body (Lemma). The upper and lower
bounds for h^0(N) and for the product's dimension are both
controlled by the volume of the jet body.
C.7. Fixed generators and neutral algebraic extensions
(Corollary). At one eventual index, the whole field generated
by a bounded-cost set lands in K_*, and extensions of
trivially valued subfields remain trivially valued.
Section D (Characters and regularization on affine cones).
D.1. Cone dictionary (Proposition). Homogeneous functions on an
affine cone give a dictionary between self-dual ray weights
and Cartier divisors on the Proj, with discrepancies and
phases expressed monomial-minimally.
D.2. All divisorial lc tests are Gauss (Lemma). Every divisorial
lc place of the cone is a Gauss valuation, so the divisorial
testing set is essentially the monomial one.
D.3. Sections as regular symbols (Lemma). Sections of the cone's
polarization correspond to regular symbols of the form
(rational function) * h^b times a monomial z^weight.
D.4. Sandwich regularization (Proposition). Given bounded phase
information on the small-discrepancy locus, a bounded
multiple of J_m has a nonzero regular symbol.
D.5. Bounded-multiple character regularization (Corollary). For
a weight in a bounded-width strip, a bounded multiple of a
character times h^b is regular.
D.6. Count for a single character and weight (Corollary). The
number of homogeneous invariant symbols in a bounded-width
character-weight box is bounded by a constant times v_i *
n^p.
D.7. Characters for a tilted height (Corollary). A bounded number
of negative-degree character eigenfunctions plus a small tilt
of the polarization gives positive-degree regularization.
Section E (Optimization on common log canonical slices).
E.1. Logarithmic jet estimate (Lemma). On a fixed resolution, the
monomial retraction and the valuation differ on polynomials of
degree at most D by O(D * discrepancy). Iterating gives
uniform control on every regular function, not only polynomials.
E.2. Compactness and uniform centring (Lemma). The common lc slice
is a compact finite polyhedral complex, with uniform lower
volume bounds on centred valuations.
E.3. Simultaneous approximation (Corollary). For a single constant
C, translation to a monomial retraction loses at most (1 + C e)
in every regular function, uniformly over all such valuations.
Thus volume of a centred valuation is controlled by the volume
of its retraction.
E.4. Convex domination (Lemma). Any convex combination of
common-slice valuations is dominated (pointwise on every
regular function) by a single common-slice valuation. The
allowed heights behave well under this domination.
E.5. Opposite ratios (Corollary). Opposite-order ratios of the
form q'/q with q*q' regular interpolate exactly under the
convex-domination step.
E.6. Volume interpolation and a strict gap (Lemma). If two
valuations differ on some regular function by more than a
fixed factor, their midpoint interpolant has strictly smaller
volume (relative to either). Near-minimizers are therefore
multiplicatively close on every regular function.
E.7. A boundary encoding the common slice (Lemma). An effective
lc boundary whose zero-discrepancy rays are exactly the
common slice can be built by adding small pencil members
with positive bounded coefficients.
E.8. Exact common-slice minimizer (Theorem). For a rational
height, minimization over the common slice is realized
exactly by the normalized-volume minimizer of an ordinary
klt pair. One single rational perturbation parameter lambda,
fixed for all competitors, suffices.
E.9. Near-minimizers for a height (Corollary). Two valuations
that are both within o(1) relative error of the minimum
must be multiplicatively 1+o(1) equal on every regular
function.
E.10. Domination by the minimizer (Corollary). Any common-slice
valuation is dominated by 2^n times the minimizer, up to
height normalization.
E.11. One-sided height tilt (Lemma). If a common-slice valuation
has a positive value on a sign function, then a small
one-sided tilt of the height yields a minimizer that also
has positive sign and stays bounded on the slice.
Section F (Degeneration and rank refinement).
F.1. Weighted presentations (Lemma). The associated graded
algebra of a quasi-monomial valuation is a quotient of a
power series ring by the initial ideal; nearby rational
regradings give nearby quasi-monomial valuations.
F.2. Canonical forms under degeneration (Proposition). The
reflexive top form descends to the central fibre with the
same grading weight and group character, and the group
action on the cone remains quasi-etale.
F.3. Lc equality and principal boundaries (Corollary). If the
degenerating valuation is h-lc, the central pair is lc and
the canonical generator has the same group character as the
initial symbol of h.
F.4. Sign transfer (Proposition). An inequality of the form
q(g) <= 0 on the actual common slice is equivalent to the
corresponding inequality on the central common slice, for
any test ratio g, without restriction on its cost.
F.5. Exact tests and pure weights (Corollary). Exact vanishing
of a test throughout the actual slice is equivalent to
exact vanishing of its initial form throughout the central
slice; pure-weight valuations are always h-lc and common.
F.6. Compression (Lemma). Two successive gradings (from two
successive minimizations) can be compressed into a single
actual valuation on the original germ, preserving all lc
equalities and all exact tests and increasing the rational
rank.
F.7. Rank refinement (Proposition). If a sequence of balanced
cones satisfies a fixed balance condition, the refinement
step replaces the valuation by a near-equal one with the
same main properties and satisfying the angular order
condition (C-alpha) with error eta_i -> 0.
Section G (Pinning the gradings).
G.1. Pinning obstruction (Theorem). An infinite sequence of
quasi-etale character-graded klt germs with lc equality and
the saturation hypothesis cannot exist. This is the
contradiction that proves the birational complement theorem.
G.2. Non-cancellation in every degree (Lemma). Among finitely
many bounded-cost monomial products, the least-w initial
term does not cancel at any later working valuation on the
pre-pinning slice, in every polynomial degree.
G.3. Uniform product growth (Lemma). For a finite-dimensional
space V of dimension g with transcendence degree a, N-fold
products of members of V have dimension at least
c_n * g * N^a.
G.4. High-scale pin (Proposition). At a high stage, finitely many
proportion tests of cost O(L) can be added to force the
selected values either to zero or to a rational line,
retaining a centred point and all previous constraints.
G.5. Field and lattice consequences of the pin (Lemma). The pin
automatically puts every lattice direction of width O(L) on
the anchor line, and forces the angular tests to act
trivially on the cutoff field at the next scale.
G.6. Additional row via tilted characters (Claim). If a
projection of the current constrained set is thin, a tilted
character block produces an integral tie at the comparator
that is transverse to the previously imposed rows.
Section H (Reduction to the pinning input).
H.1. Factorial saturation (Proposition). Failure of klt
complements gives a factorial order w whose section orders
dominate k A_X(w) on every prime, uniformly, and reduces
to a geometric generic slice.
H.2. The isomorphic case (Lemma). In the birational-but-Cartier
case, a canonical cover produces the exact pinning input
(torsion, character match, lc equality, saturation).
H.3. The signed anti-canonical torsor (Lemma). In the general
birational case, a signed anti-canonical Z-graded torsor
serves as the model, with both positive and negative slopes
to catch section vanishing.
H.4. Padding and the actual quotient discrepancy (Lemma).
Padding by b-1 extra coordinates makes the canonical
character match b and preserves a fixed quotient
discrepancy gap e.
H.5. The pinning input (Proposition). The reduced data satisfy
every hypothesis of Theorem G.1 (the pinning obstruction).
H.6. Point ideals and change of field (Lemma). Reduction to C
via descent, and the passage between the lc threshold of
the point ideal and existence of a good Cartier divisor
through the point.
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