FUJITA'S FREENESS CONJECTURE
A complete proof, written to be read from top to bottom
CONTENTS
0. What the theorem says, and why n+1 is exact
1. A dictionary of the words used
2. The plan in five steps
3. Foundations: orders, discrepancies, thresholds, vanishing
4. Step one: failure at a point forces a strict order inequality
5. Step two: the best test exists
6. Step three: every best test is centered at the point
7. Step four and five: isolate one divisor, lift a nonzero value
8. What is used where
========================================================================
0. WHAT THE THEOREM SAYS, AND WHY n+1 IS EXACTLY RIGHT
========================================================================
Throughout, X is a smooth connected projective variety over the complex
numbers C, of dimension n at least 1, and L is an ample line bundle on X.
The adjoint bundle is P = K_X + (n+1)L, where K_X is the canonical
bundle and + means tensor product (equivalently, addition of divisors).
THEOREM (the Fujita freeness conjecture).
K_X + (n+1)L is generated by its global sections.
Equivalently: for every point x of X there is a global section s of
K_X + (n+1)L with s(x) not equal to 0. That is what globally generated
(also called spanned, or free) means.
Connectedness is bookkeeping only: the statement applies to one
connected component at a time. In dimension zero (a single point)
every line bundle is generated.
COROLLARY (all higher powers). K_X + mL is globally generated for every
integer m >= n+1. Proof: put r = m-(n+1) and apply the theorem to the
product X x P^r with the ample bundle L tensor O(1) on the P^r factor.
The adjoint bundle there restricts to K_X + mL on each slice X x point,
and a restriction of a globally generated bundle is globally generated.
For r = 0 this is the theorem itself.
WHY THE BOUND n+1 IS SHARP. Take X = P^n and L = O(1). Then
K_X = O(-(n+1)), hence K_X+(n+1)L = O(0), the trivial bundle, which has
the constant section 1, nowhere zero: globally generated. But
K_X+nL = O(-1) has no nonzero sections at all. So the threshold is
exactly n+1, and no smaller uniform twist works for all (X,L).
WHY THE THEOREM IS NOT EASY. On P^n the canonical bundle is very
negative, yet the twist repairs it. On a general X, K_X can be
arbitrarily negative, X can be high-dimensional, and an ample L might
itself have almost no sections. The hard part is to find ONE section
that does not vanish at a PRESCRIBED point x, uniformly over all shapes
and dimensions of X and all ample L. The proof converts failure at x
into a sharp numerical contradiction, then lifts a nonzero value.
Historical landmarks (orientation only, not used below): curves by
Riemann-Roch; surfaces by Reider; threefolds by Ein-Lazarsfeld;
fourfolds by Kawamata; fivefolds by Ye-Zhu. In arbitrary dimension,
Angehrn-Siu gave a quadratic bound, then Helmke, Heier, Ghidelli-Lacini
and Han improved toward linear. The proof below reaches n+1 exactly.
========================================================================
1. A DICTIONARY OF THE WORDS USED
========================================================================
Read the proof with this section as background; return to it as needed.
projective variety: the common zero set of finitely many homogeneous
polynomials in some projective space, with its algebraic structure.
It is compact. Smooth means no corners or self-touching points: near
each point it looks like an open ball in C^n.
dimension n: the number of independent complex directions at a smooth
point; C^n has dimension n.
line bundle: a way to attach a one-dimensional complex vector space to
every point, varying algebraically. Two models: (a) a locally trivial
family of lines glued by transition functions; (b) the data of a
divisor D up to linear equivalence, written O(D). Tensor product is
written additively, O(D)+O(E)=O(D+E); the dual is O(-D).
ample: positive enough. Equivalently some positive power L^k gives an
embedding of X into a projective space (L^k is very ample).
section of a line bundle: a rule choosing a vector in the line over each
point, varying algebraically. The zero set of a nonzero section is a
divisor. H^0(X,L) is the vector space of all sections; its dimension is
h^0(X,L).
globally generated (spanned, free): for every point x there is a section
not vanishing at x. Equivalently the sections define a regular map
X -> projective space, with no indeterminacy.
canonical bundle K_X: the top exterior power of the cotangent bundle;
concretely the bundle of holomorphic n-forms. It can be very negative,
as on P^n.
adjoint bundle: K_X + mL. The conjecture is that m = n+1 always gives
global generation.
divisor: a formal finite integer combination of codimension-one
subvarieties. Effective means all coefficients are >= 0.
log resolution: a birational map pi: Y -> X from a smooth Y on which the
preimage of a given divisor, plus all exceptional divisors, is a simple
normal crossing (SNC) divisor: locally a union of coordinate
hyperplanes. Existence in characteristic zero is the Hironaka theorem.
exceptional divisor: a prime divisor on Y mapping to a positive-
codimension subset of X (it does not survive as a divisor downstairs).
nef and big: nef means nonnegative on every curve; big means top self-
intersection grows like an ample bundle. Nef and big is positive enough
for vanishing theorems.
Kawamata-Viehweg vanishing: the cohomological engine, stated in Section
3. It turns positivity into vanishing of higher cohomology, which lets
a section on a subvariety lift to the whole space.
========================================================================
2. THE PLAN OF THE PROOF IN FIVE STEPS
========================================================================
We argue by contradiction. Suppose P = K_X + (n+1)L is NOT generated at
some point x. We derive a contradiction in five steps; each is fully
proved later, and Section 8 lists exactly what is used where.
STEP ONE (Section 4). Translate fails-at-x into a strict inequality on
how fast sections of large mL vanish at x. Take a basis of H^0(X,mL),
read off each section Taylor initial exponent at x, and average the
total degrees. The normalized average has liminf strictly larger than
n/(n+1). This is a clean statement about lattice points.
STEP TWO (Section 5). Set up a minimization. Among all bases of
H^0(X,mL) and all monomial tests v (weighted orders of vanishing with a
log discrepancy A(v)), minimize the exponential average
(1/N_m) sum_i exp( A(v)/t - v(s_i)/m ), for a fixed t < n+1.
Show the minimum is attained and that the discrepancy of every
minimizing test lies in a fixed compact interval [a_star, a^*] with
a_star > 0, uniformly in m and t. This uniformity is the technical
heart of making the best test well defined.
STEP THREE (Section 6). Show every minimizer is centered exactly at the
point x. If a minimizing test had a positive-dimensional center W, a
variation tangent to W would force a weighted tangential moment to be at
most d/t, while a uniform restriction estimate plus a discrete
Brunn-Minkowski inequality would force the same moment to be at least
d/t with a strict margin. Contradiction.
STEP FOUR (Section 7). A point-centered minimizer gives a rational
supporting functional on finitely many polytopes of normalized section
orders. From it we build an effective ideal and an SNC boundary B with a
reduced divisor S over x on which the coefficient of B equals the log
discrepancy, while every other divisor meeting S gets a strictly
smaller coefficient.
STEP FIVE (Section 7). Kawamata-Viehweg vanishing makes the restriction
map H^0(T,O(D+S)) -> H^0(S,O_S(D+S)) surjective. A nonzero value in the
fiber P|_x gives a nonzero section of O_S(D+S); lift it and descend to a
section of P not vanishing at x. Contradiction.
========================================================================
3. FOUNDATIONS: ORDERS, DISCREPANCIES, THRESHOLDS, VANISHING
========================================================================
Notation throughout the proof. X, L, n are as in the theorem. Put
P = K_X + (n+1)L, R_j = H^0(X, jL), N_j = dim R_j.
Ampleness gives the Hilbert asymptotic
N_j = (L^n / n!) j^n + O(j^(n-1)), with L^n a positive integer.
All centers are CLOSED centers. So a divisor centered along a
subvariety W is an admissible test at every point of W, not just at its
generic point.
WEIGHTED MONOMIAL ORDERS. Fix a smooth projective birational model
pi: Y -> X and local regular coordinates adapted to the ENTIRE relative
canonical divisor: every component of that divisor meeting the
coordinate neighborhood is a coordinate hyperplane. Choose a subset
E_1,...,E_r of the coordinate divisors and give them positive real
weights u_1,...,u_r; all remaining coordinates get weight zero. The
order of a germ is the least weight of a nonzero monomial in the local
equations of the E_j, with coefficients in the remaining coordinates.
The log discrepancy of this test is
A(v) = sum_j u_j A(E_j),
where for a prime divisor F on a model, A(F) = 1 + ord_F(K_{Y/X}) is
the log discrepancy. A coordinate divisor absent from the relative
canonical divisor has A = 1. The chosen test has a positive-support
stratum; we use its closed image on X as the center. This adaptation
condition is exactly what makes the displayed formula for A(v) correct.
Positive multiples of a prime order are allowed, and scaling discrepancy
by the same factor. Approximating real weights by positive rational
weights on a fixed support approximates both the discrepancy and the
orders of any fixed finite family of sections.
LEMMA (local computation of a monomial order). A monomial order with
fixed positive support can be computed in the Taylor ring at any smooth
point of its open stratum. Its value is unchanged by replacing each
positive-weight coordinate by a unit multiple, or by changing the
complementary zero-weight coordinates.
Proof. For a threshold c, the relevant ideal is generated by monomials
in the positive-weight parameters of weight at least c, and these are
preserved up to units. In the completed local ring the ideal is
extended from a finite-colength monomial ideal in the positive-weight
variables, hence primary to their intersection. Localization at the
generic point of the stratum followed by contraction does not change
membership, and completion is faithfully flat. So a coefficient that
vanishes at the chosen point but not identically on the stratum is a
nonzero series in the zero-weight variables; it does not raise the
weighted order.
LEMMA (SNC retraction). Let D_1,...,D_N be effective divisors on X and
let Y -> X be a log resolution of their union, with SNC exceptional
components E_j. For a positive multiple v of a prime-divisor order over
Y set u_j = v(E_j). Then
v(D_i) = sum_j u_j ord_{E_j}(D_i), and A(v) >= sum_j u_j A(E_j).
If some u_j is nonzero, the monomial order with these weights at the
stratum containing the generic center of v has the same orders on the
D_i, no larger discrepancy, and a closed center on X containing that of
v.
Proof. Each pulled-back divisor is locally a monomial times a unit,
giving the first equality. The reduced SNC divisor is log canonical: a
logarithmic volume form has at most a simple pole along each prime on a
higher model. So the log discrepancy relative to Y is at least sum u_j;
adding the order of K_{Y/X} gives the inequality and the claimed
discrepancy of the monomial order. The stratum closure contains the
generic center, so its image contains the original closed center.
THRESHOLDS. For a coherent ideal a on a smooth variety, the local log
canonical threshold at x is computed on a log resolution by
lct_x(a) = min over F with x in pi(F) and ord_F(a) > 0 of
A(F) / ord_F(a).
The minimum is +infinity for the unit ideal, and 0 for the zero ideal.
LEMMA (closed threshold loci). Let V be a smooth complex projective
variety, S a complex algebraic variety, and a an ideal on V x S given
locally by algebraic families of generators. For a real c, the locus
{(y,s) in V x S : lct_y(a_s) <= c}
is Zariski closed, where a_s is the ideal generated by the restricted
generators (no flatness required).
Proof. The analytic semi-continuity of complex singularity exponents of
Demailly-Kollar gives closedness in the classical topology, because
phi = (1/2) log sum |g_i|^2 has e^phi locally Holder continuous and
translation handles a moving marked point. Constructibility comes from
stratifying: over the generic point of each component resolve the ideal,
spread the resolution over a dense open, and the fiber maps are log
resolutions with fixed orders and coefficients, so the threshold
condition is a finite union of proper images (algebraic, proper, hence
constructible); induction on the complement finishes. A subset that is
constructible and classically closed is Zariski closed, since a Zariski
open subset of an irreducible variety is classically dense.
THEOREM (Kawamata-Viehweg vanishing). Let T be smooth projective over C.
Let Delta be an effective rational SNC divisor with coefficients in
[0,1), and let D be an integral divisor. If D - (K_T + Delta) is nef and
big, then H^i(T, O_T(D)) = 0 for every i > 0.
Equivalently, using Q = D - K_T - Delta and its rounding: if M is
integral and B is an effective rational SNC divisor with M - B nef and
big, then H^i(T, O_T(K_T + M - floor(B))) = 0 for i > 0. This is the
form used below.
LEMMA (descent). Let pi: T -> V be a proper birational morphism of
smooth varieties, P_V a line bundle on V, and C an integral divisor on T
whose positive part is exceptional. A section of pi^*P_V + C defines a
regular section of P_V on V. If a prime F on T maps to a point x, the
coefficient of C on F is zero, and the section is nonzero generically on
F, then the descended section is nonzero at x.
Proof. View the section as a rational section of P_V on the common
function field. Its orders at all nonexceptional primes are
nonnegative, because the coefficients of C there are nonpositive. So it
is regular in codimension one on the normal variety V, hence
everywhere. Under the extra assumptions its pullback has order zero
along F. A local function vanishing at x has positive order along every
prime centered at x, so the descended section cannot vanish at x.
========================================================================
4. STEP ONE: FAILURE AT A POINT FORCES A STRICT ORDER INEQUALITY
========================================================================
Fix a closed point x in X. Choose regular parameters at x and a local
frame of L; use powers of that frame as frames of jL. Order Taylor
monomials first by total degree, then lexicographically, always taking
the smallest monomial. Let
H_j = { in(s) : 0 != s in R_j } inside Z_{>=0}^n
be the set of initial exponents. Then |in(s)|_1 = mult_x(s), the
multiplicity (order of vanishing) of s at x.
LEMMA (initial exponents give a basis). The set H_j has cardinality N_j,
and representatives of its distinct exponents form a basis of R_j.
There is a constant C > 0, independent of j and s, with
mult_x(s) <= C j for every nonzero s in R_j.
Moreover H_j + H_q is contained in H_{j+q}.
Proof. Taylor expansion is injective on R_j: a section with zero germ
vanishes on a nonempty open set, hence on the integral variety X. Each
monomial coefficient ranges over a one-dimensional space. Successively
select a smallest initial monomial and eliminate its coefficient from
the remaining vectors; this produces a basis with distinct initials, and
any linear combination has the initial of its first nonzero summand.
That proves the cardinality claim. For the multiplicity bound, choose a
fixed sufficiently ample multiple kL, and general members of its linear
system through x cut out a curve smooth at x and not contained in the
divisor of a prescribed nonzero section of jL. Local intersection
multiplicity bounds mult_x(s) by the degree of that divisor on the
curve, at most j k^(n-1) L^n. In dimension one this is just the degree
bound for an effective divisor on X. Finally the monomial order is
multiplicative, so initial exponents add under products of sections.
PROPOSITION (ordinary-order alternative). If P = K_X+(n+1)L is not
generated at x, then
liminf over j of (1/(j N_j)) * sum_{gamma in H_j} |gamma|_1
> n/(n+1).
Proof. We prove the contrapositive: failure of the strict inequality
forces generation at x. Three steps: lattice near-equality forces
L^n = 1 along a subsequence; the leading forms on the exceptional
divisor of the blowup at x then have a strict multiplier threshold;
vanishing lifts a nonzero section from that divisor.
Put V = L^n. The number of nonnegative integral points of total degree
at most r is binom(r+n, n). Filling points in order of increasing total
degree, and using N_j = V j^n / n! + O(j^(n-1)), gives
liminf (1/(j N_j)) sum_{gamma in H_j} |gamma|_1
>= V^(1/n) * n/(n+1).
Indeed the last occupied degree for the N_j smallest lattice points is
V^(1/n) j + O(1), and summing total degree over these points gives mean
degree V^(1/n) n j/(n+1) + O(1). Since V is a positive integer, V^(1/n)
is at least 1, so if the strict inequality of the proposition fails,
then necessarily V = 1 and equality holds along a subsequence.
Let Delta = {u in R^n : u >= 0, |u|_1 <= 1}. Along the equality
subsequence, every fixed open ball with closure in the interior of Delta
meets j^{-1} H_j for all large j. (Compare H_j with the N_j smallest
lattice points T_j. If such a ball were missed, at least c_1 j^n points
of T_j of degrees at most (1-c_2) j would be absent; each replacement
outside T_j has degree at least j-O(1); the remaining replacements
cannot lower the degree sum, so the normalized mean would exceed the
minimum by a fixed amount, contradicting equality.)
Choose a rational t with n < t < n+1 and put c = (1/t,...,1/t), a
point in the interior of Delta. Choose finitely many small balls with
closures in that interior such that any choice of one point per ball has
convex hull containing c in its interior. For one large degree j on the
subsequence, choose initial exponents gamma_nu in H_j, one per ball after
scaling by 1/j. The rational polytope Q = conv{gamma_nu / j} contains c
in its interior. Its slice {|u|_1 = n/t} has rational points p_1,...,p_b
whose convex hull contains c in its relative interior (in dimension one
that slice is the single point c).
Write each p_l as a rational convex combination of the gamma_nu / j and
choose sections s_nu with those initials. If t = a/b in lowest terms,
choose Q_0 divisible by a and by every denominator; then products of Q_0
sections give sections r_1,...,r_b of j Q_0 L. Put h = j Q_0 / t, a
positive integer by construction. Their initial exponents beta_l satisfy
r_l in H^0(X, h t L), beta_l = j Q_0 p_l, |beta_l|_1 = n h.
Thus conv{beta_l / h} contains the all-ones vector 1 in relative
interior inside the hyperplane {|u|_1 = n}.
Let p: B = Bl_x X -> X be the blowup, E the exceptional divisor
(P^{n-1} for n>1; for n=1 take p the identity and E = x). Removing
the common order n h from the pullbacks of the r_l gives a subsystem of
H = p^*(h t L) - n h E
with base ideal b. Its restriction to E is generated, in local frames,
by the leading homogeneous forms g_l of the r_l, each of degree n h.
CLAIM. lct_z(E, b|_E) > 1/h for every z in E. (For n=1 the restricted
ideal is the unit ideal.)
For n > 1, replace the g_l by their initial monomials z^{beta_l}. On the
chart of E where z_k != 0 their exponents are obtained by omitting
coordinate k. Projection from the hyperplane |u|_1 = n is an affine
isomorphism, so the normalized exponent polytope contains the all-ones
vector in ordinary interior. Hence for every nonzero u in R_{>=0}^{n-1},
(1/h) min_l <u, beta_l^(k)> < sum_i u_i.
For a prime order over this chart use its values on the coordinates as
u. The left side is the order of the monomial ideal over h; the right
side is at most the log discrepancy (SNC inequality of the retraction
lemma). So the monomial ideal is klt at coefficient 1/h; computing its
threshold on a log resolution gives a strict threshold > 1/h at every
point. (If u=0 the order is 0 and the discrepancy is positive.)
Transfer to the original forms by an algebraic degeneration. Only
finitely many monomials occur; choose an integral weight w making the
lex initial of each g_l its unique smallest-weight term. The forms
g_{l,lambda}(z) = lambda^{-<w,beta_l>} g_l(lambda^{w_1} z_1, ...)
have polynomial coefficients in lambda with monomial special fiber.
They define an algebraic family of ideals on E x A^1. By the closed-
threshold-loci lemma, the locus where the fiber threshold is <= 1/h is
closed; its image in A^1 is closed (E is proper) and misses 0. For some
nonzero parameter the threshold exceeds 1/h everywhere on E. Every
nonzero fiber is a diagonal automorph of the original forms, so the
claim holds for the g_l.
It remains to lift from E. The blowup formula gives
p^*P - (K_B + E) - H/h = (n+1-t) p^*L.
Take a projective log resolution sigma: T -> B of E and b, with strict
transform Eprime and b O_T = O_T(-G). Eprime is not a component of G
(the restricted system is not identically zero). Set
D_T = K_T + sigma^*(p^*P - K_B - E) - floor(G/h),
an integral divisor. Then
D_T - K_T - {G/h} = (n+1-t)(p sigma)^*L + (sigma^*H - G)/h.
The first summand is nef and big; the second is nef, since sigma^*H - G
is globally generated. The fractional boundary has SNC support. By
Kawamata-Viehweg vanishing, H^1(T, O_T(D_T)) = 0.
Let sigma_E: Eprime -> E be the induced morphism. Adjunction identifies
the restriction of D_T + Eprime with
sigma_E^* (p^*P |_E) + K_{Eprime/E} - floor(G|_{Eprime}/h).
Indeed Eprime + Supp(G) is SNC, each component restricts with
multiplicity one, distinct components cannot restrict to the same prime,
so taking floors commutes with restriction; and the generators on
Eprime generate O_{Eprime}(-G|_{Eprime}), the pullback of the
restricted ideal. The strict threshold implies
K_{Eprime/E} - floor(G|_{Eprime}/h) >= 0,
an effective correction. The first bundle is trivial with fiber P_x. A
nonzero vector in that fiber, times the canonical section of the
effective correction, gives a section on Eprime nonzero at its generic
point. The exact sequence for Eprime and the vanishing lift it to a
section of D_T + Eprime.
Relative to sigma^* p^* P, the correction of this lifted bundle is
C_T = K_{T/B} - sigma^* E + Eprime - floor(G/h).
Its coefficient on Eprime is zero, and on every other sigma-
nonexceptional divisor it is nonpositive; its positive part is thus
exceptional over X. Since Eprime maps to x and the lifted section is
nonzero at its generic point, the descent lemma applied to p sigma gives
a section of P nonzero at x. This proves the proposition, including
dimension one, when Eprime is the point itself. END OF PROOF.
========================================================================
5. STEP TWO: THE BEST TEST EXISTS (EXPONENTIAL MINIMIZATION)
========================================================================
We now let both the basis and the test vary, and minimize
F_{m,t}(v, s) = (1/N_m) sum_i exp( A(v)/t - v(s_i)/m ), t > 0,
over ordered bases s = (s_1,...,s_N_m) of R_m and monomial tests v. Let
f_{m,t}(z) = inf over v with z in c(v), and over bases, of F_{m,t}.
Here c(v) is the closed center. Positive rescaling of v is allowed:
approximating its positive weights by rational weights preserves the
center and approximates the orders of finitely many sections and the
discrepancy. So the same infimum is obtained using positive multiples of
prime-divisor orders.
LEMMA (order bound). If the center of v contains x, then every nonzero
section s satisfies
v(s) <= A(v) mult_x(s).
In particular, for s in R_m, 0 <= v(s)/A(v) <= C m, with C from the
initial-exponent lemma.
Proof. For an effective divisor with local equation g of multiplicity
r > 0 at a smooth point, lct_x(g) >= 1/r. (Weierstrass preparation: on
a small polydisc g is a unit times a monic degree-r polynomial in one
coordinate; for any c < 1/r Holder bounds the integral of |g|^{-2c} by
products of finite one-variable integrals, so Fubini gives local
integrability.) Computing the threshold on a log resolution gives the
bound v(s) <= A(v) mult_x(s) for prime orders whose center contains x,
and rational approximation gives it for monomial tests.
LEMMA (uniform sublevel bounds). Suppose P is not generated at x. Then
there are numbers 0 < t_0 < n+1, c > 0, 0 < delta < 1, 0 < a_star <=
a^* < infinity, and an integer m_0 such that for every m >= m_0 and
every t in [t_0, n+1],
c <= f_{m,t}(x) <= 1 - delta.
Moreover any pair centered with image containing x and with
F_{m,t}(v,s) <= 1 - delta/2 satisfies a_star <= A(v) <= a^*. All these
constants are independent of m and t in the stated ranges.
Proof. By the ordinary-order alternative there is epsilon > 0 with
(1/(m N_m)) sum_{gamma in H_m} |gamma|_1 >= n/(n+1) + 3 epsilon
for all large m. Choose t_0 < n+1 so close that n/t_0 <=
n/(n+1) + epsilon. Take a basis with distinct ordinary initials and use
v = lambda * ord_x, the ordinary point order, with A(ord_x) = n (for
n=1 this is the order of the point). The linear coefficient at
lambda = 0 is at most -2 epsilon uniformly in m and t; by the
multiplicity bound the quantities n/t - mult_x(s_i)/m are uniformly
bounded, so the quadratic remainder is O(lambda^2). A fixed small
lambda gives F_{m,t} <= 1 - delta for a fixed delta > 0.
Next a uniform supply of low-order jets. Choose k with kL very ample
and sections s_0,...,s_n of kL with s_0(x) != 0 and z_i = s_i/s_0
regular parameters. For each residue class mod k choose a fixed large
b_r in that class with b_r L globally generated and tau_r in
H^0(X, b_r L) nonzero at x. Write m = qk + b_r. For |alpha|_1 <= q the
sections tau_r s_0^{q-|alpha|} s_1^{alpha_1}...s_n^{alpha_n} have germs
equal to a common unit times z^alpha in a local frame of mL. Let
r_m = min(q, floor(m/(2(n+1)))).
Their jets for |alpha|_1 <= r_m are linearly independent modulo order
> m/(2t). For large m, r_m is at least a fixed positive multiple of m,
so Hilbert asymptotics give a constant 0 < kappa < 1 with
codim { s in R_m : mult_x(s) > m/(2t) } >= kappa N_m.
By the order bound the subspace { s : v(s) > m a/(2t) } sits inside
this one, where a = A(v). Thus at least kappa N_m vectors of any basis
(adapted or not) have v(s_i) <= m a/(2t), so
F_{m,t}(v,s) >= kappa exp( a/(2t) ) >= kappa exp( a/(2(n+1)) ).
This gives the lower bound c = kappa and bounds a above on the sublevel
set. Finally F >= exp(-C a) by the order bound and multiplicity bound;
choosing a_star small enough that exp(-C a_star) > 1 - delta/2, no
sublevel pair has a < a_star. Enlarging the upper bound if needed gives
a^* >= a_star.
LEMMA (closed realizability loci). Fix m with N_m > 0 and a decreasing
list alpha_1 >= ... >= alpha_{N_m} >= 0 of rationals. The set of points
z in X for which some ordered basis and some prime-divisor order v with
z in c(v) satisfy v(s_i)/A(v) >= alpha_i (all i) is Zariski closed.
There is also a closed incidence locus in the product of X with the
complete flag variety of R_m, recording realizability by bases adapted
to a given flag.
Proof. Let F be the projective flag variety; a flag has subspaces
U_i of dimensions i and base ideals b_i, by evaluation after twisting by
-mL. If the list holds, the flag spanned by the first i basis vectors
satisfies v(b_i) >= alpha_i A(v) (because the list is decreasing);
conversely these inequalities imply the list for every adapted basis.
If some alpha_i > 0, choose M making M/alpha_i integral for each
positive entry and set a_alpha = sum_{alpha_i>0} b_i^{M/alpha_i}. The
simultaneous inequalities are equivalent to v(a_alpha) >= M A(v). A
divisorial witness with center containing z exists iff
lct_z(X, a_alpha) <= 1/M,
since the local threshold is computed by a component on a log
resolution. The universal subbundles on F give algebraic families of
generators on X x F, hence for the powers and sum; the closed-
threshold-loci lemma applied to this condition gives a closed incidence
locus, whose projection to X is closed (F proper). If all entries
vanish, both loci are the whole parameter spaces.
PROPOSITION (existence and uniform bounds for minimizers). Suppose P is
not generated at x. For the constants and ranges of the coercivity
lemma, f_{m,t}(x) satisfies c <= f <= 1-delta and is attained by an
ordered basis of R_m and a nonzero real-weight monomial test on a
projective log resolution of the basis divisors. Every minimizing test
has a_star <= A(v) <= a^*. The same bounds hold eventually along every
minimizing sequence, so a_star is uniformly positive.
Proof. Only attainment remains. Fix m,t in range and take divisorial
pairs whose objective decreases to f_{m,t}(x); eventually they lie in
the sublevel set. Order each basis by decreasing normalized values and
pass to a subsequence with A(v_b) -> a in [a_star, a^*] and
q_{b,i} = v_b(s_{b,i})/A(v_b) -> q_i. This is possible since
0 <= q_{b,i} <= C m. The limit list is decreasing and
f_{m,t}(x) = (1/N_m) sum_i exp( a/t - a q_i/m ).
At least one q_i is positive, else the right side is exp(a/t) > 1,
contradicting f <= 1 - delta.
Choose decreasing nonnegative rational lists alpha^(k) increasing
componentwise to (q_i), strictly below q_i whenever q_i > 0. Each list
is satisfied by all sufficiently late pairs. By the closed-realizability
lemma the flags admitting a witness at x for alpha^(k) form a nonempty
closed subset F_k of the flag variety; these are nested, so projectivity
gives a flag in their intersection. Fix a basis s_1,...,s_{N_m} adapted
to this flag. Resolve the divisors of these sections on one smooth
projective model Y -> X, including exceptional divisors in the SNC
support. For each large k, choose a witness for the fixed flag and
alpha^(k) on a common smooth model dominating Y. Retracting to the fixed
SNC model (retraction lemma) preserves all section orders, does not
increase discrepancy, and has center on X containing x. Normalize to
discrepancy one; it satisfies tilde_v_k(s_i) >= alpha_i^(k).
There are finitely many SNC strata and components on Y. Pass to a
subsequence using the same stratum component, coordinate divisors indexed
by a fixed set J. Their normalized weights belong to the compact simplex
{ (u_j) : u_j >= 0, sum_j A(E_j) u_j = 1 }.
All A(E_j) are positive since X is smooth. Pass to a limit of the
weights; some u_j may become zero. Take the component of the remaining
positive-support intersection that contains the old stratum component;
its closure contains the old center, so its image still contains x. The
limit is a nonzero monomial test tilde_v of discrepancy one, and section
orders are linear in the weights, so tilde_v(s_i) >= q_i for all i.
Set v = a tilde_v. Comparing with the limit objective,
F_{m,t}(v,s) <= (1/N_m) sum_i exp( a/t - a q_i/m ) = f_{m,t}(x).
The reverse inequality holds by definition of the infimum, so this pair
attains it. The discrepancy bounds for all minimizers and eventually for
minimizing sequences follow from the uniform sublevel bounds. END PROOF.
LEMMA (specialization comparison). Fix m with N_m > 0 and a closed
irreducible W in X containing x. At a very general closed point y of W,
for every t > 0,
f_{m,t}(y) >= f_{m,t}(x).
Proof. There are countably many rational lists as in the closed-
realizability lemma. For each, intersect its closed locus with W and
discard it if the intersection is proper in W; the complement contains
very general closed points. For any remaining y, every such list
realized at y is realized at x. Take a test and basis at y, reorder so
q_i = v(s_i)/A(v) is decreasing, a = A(v) > 0. Approximate (q_i) from
below by rational lists; each is realized at x, and after scaling to
discrepancy a gives an objective at x at most
(1/N_m) sum_i exp( a/t - a alpha_i^(b)/m ).
Taking the limit gives f_{m,t}(x) <= F_{m,t}(v,s). Infimizing over tests
at y proves the claim. The discarded loci are independent of t, giving
the simultaneous statement for all positive t.
========================================================================
6. STEP THREE: EVERY BEST TEST IS CENTERED AT THE POINT
========================================================================
PROPOSITION (point center). Suppose P is not generated at x. Then there
is a real 0 < t < n+1 such that, for arbitrarily large integers m, the
infimum f_{m,t}(x) is attained and EVERY minimizing monomial pair has
center exactly {x}. (A minimizing pair is a monomial test together with a
basis attaining the infimum, over all such pairs.) We keep this one t and
this point x fixed from here on.
Before the argument, a uniform estimate: the image of the restriction
map from sections of hL to a subvariety W has dimension bounded below by
a function of h alone, uniform over all W of a fixed dimension.
Fix k with kL very ample and embed X by its complete system. For an
integral d-dimensional closed W in X put
I_W(h) = dim image( R_h -> H^0(W, hL|_W) ).
LEMMA (uniform restriction estimate). For each 1 <= d <= n there are
constants C_d, H_d depending only on (X,L,k,d) with
(d! I_W(h))^{1/d} >= h - C_d for all h >= H_d,
for EVERY integral d-dimensional closed W in X.
Proof. Set D_0 = k^d. The degree of W in the fixed embedding is
D = (kL)^d . W = k^d (L^d . W) >= D_0, since L^d . W is a positive
integer. A general linear projection gives a finite morphism
pi: W -> P^d of degree D, with separable generic fiber (characteristic
zero). Write K = C(P^d), E = C(W). Choose a nonzero linear coordinate
s_0 of the projection; all points of the geometric generic fiber lie in
s_0 != 0. Select D_0 distinct such points. Over an algebraic closure of
K, affine linear forms separate any two of them; for each selected
point, multiplying at most D_0 - 1 such forms gives a polynomial
vanishing at the others and not at that one. Homogenizing with s_0 gives
ambient degree-D_0 forms whose normalized values span the functions on
these D_0 points.
Let V_{D_0} be the space of ambient degree-D_0 forms. Restriction and
division by s_0^{D_0} give a K-linear map V_{D_0} (x) K -> E of rank at
least D_0 (check rank after extending scalars, where the evaluation
argument applies). Choose ambient degree-D_0 forms F_1,...,F_{D_0} whose
normalized images F_i/s_0^{D_0} are K-independent; they can be chosen
among a fixed monomial spanning set, their coefficients not depending on
elements of K. For q >= D_0, multiply each F_i by a monomial basis of
forms of degree q - D_0 in the projection coordinates. The products
restrict to independent sections on W (a relation, divided by s_0^q,
would contradict K-independence; the base forms then vanish on P^d).
They are restrictions of sections in R_{qk}, so
I_W(qk) >= D_0 binom(q - D_0 + d, d),
hence (d! I_W(qk))^{1/d} >= k(q - D_0) = qk - k^{d+1}. To cover all
degrees, choose once and for all an integer h_r in each residue class r
mod k with h_r L globally generated; there is a section of h_r L not
identically zero on any W, and multiplication by it injects the
degree-qk restriction space into the degree-(qk + h_r) one (W integral).
For h = qk + h_r, q >= D_0, this gives (d! I_W(h))^{1/d} >=
h - (h_r + k^{d+1}). Taking maxima over the finitely many residue
classes proves the lemma, with both C_d and H_d at most
max_r h_r + k^{d+1}, independent of W.