Minimal models and Mori fibre spaces for generalized log canonical Q-pairs

Proof. Apply Theorem 1.1 to the fixed zero nef b-divisor.□

Proof. The assertion is immediate in dimension zero; in dimension one a smooth projective curve with pseudo-effective canonical divisor already has nef canonical divisor. Suppose that n=\dim X\geq2. Such an H is obtained from sufficiently positive general divisors with small coefficients. The ordinary klt MMP, including existence of flips [11], gives a program

with X_i projective, \Q-factorial and klt. Write H_i for the strict transform of H and \lambda_i for the scaling number at its ith step. If the program is infinite, the \lambda_i lie in (0,1], are nonincreasing and rational: a contracted curve is K_{X_i}-negative and (K_{X_i}+\lambda_iH_i)-trivial. Only finitely many steps can be divisorial, because each such step decreases the Picard number.

For a divisorial valuation u of the common function field, set

We use compatible canonical divisors on every common resolution. At step j the pullbacks of the adjoints K+\lambda_jH agree. Their difference is exceptional over the output and numerically trivial there, so this follows from the negativity lemma. The difference of the canonical pullbacks is effective by the same lemma. It follows, by summing the stepwise identities, that for 0<t\leq\lambda_i the difference between the pullback of K_X+tH and that of K_{X_i}+tH_i is effective and exceptional over X_i, and its value at u is

Thus the transformed pair (X_i,tH_i) is klt. For rational t, this effective exceptional difference identifies the systems of sufficiently divisible multiples on the two sides after adding their fixed part. In particular, K_{X_i}+tH_i is big. At t=\lambda_i it is also nef, and klt base point freeness makes it semiample.

We spell out why

Otherwise choose j after the last divisorial contraction, put S=X_j, and choose a positive rational \beta<\lim_i\lambda_i. The tail maps from S are small. For \beta\leq t\leq\lambda_j, the boundaries tH_j are klt and their adjoints are big. Moreover, H_j is big: sections of sufficiently divisible multiples of H push to sections of its transform. Choose a decomposition

where G\geq0 and the A_a are general, small effective ample rational divisors whose numerical classes span N^1(S)_\R. In each tH_j replace only a fixed sufficiently small rational portion \delta H_j, with 0<\delta<\beta, by this expression. The resulting boundaries remain klt, since the entire compact interval can be checked on one log resolution. Small variations of the coefficients of the A_a give a rational polytope of klt boundaries with a common ample part and big adjoints. BCHM finiteness of marked ample models [11, Corollary 1.1.5] applies to this polytope.

Small \Q-factorial modifications identify the numerical divisor spaces. For completeness, if a divisor is numerically trivial on one model, its pullback and the pullback of its transform differ by a divisor exceptional over the other model; that difference is numerically trivial over that model and is zero by negativity. Hence numerical equivalence is transported in both directions. At each scaling value \lambda_i, the nef class K_{X_i}+\lambda_iH_i can therefore be made ample by an arbitrarily small rational perturbation in the displayed spanning directions. The perturbation may be chosen within the fixed polytope. Smallness identifies the corresponding section systems, so the marked model S\dashrightarrow X_i is one of its finitely many ample models. No marked model can recur: discrepancies are nondecreasing at each flip, and some discrepancy increases strictly. Indeed, otherwise the canonical pullbacks would agree, contradicting the negative and positive intersection signs on the two sides of the flip over its contraction base. This proves (3.4).

For rational 0<t\leq1, put L_t=K_X+tH. Its asymptotic base order is

where m runs through sufficiently divisible positive integers. The values \sigma_u(t) increase as t decreases: a sufficiently divisible multiple of (t'-t)H is globally generated when t'>t. At a scaling value there is no asymptotic base order on X_i, by semiampleness, so (3.3) gives

The second inequality follows by fixing i, observing that \sigma_u(t)\geq F_i(t;u) for 0<t\leq\lambda_i, and letting t tend to zero.

A uniform multiplier-ideal estimate

For each positive integer s, consider

These ideals form a decreasing family as t\downarrow0, by the same comparison of base ideals. They stabilize to an ideal \mathcal J_s. Here stabilization follows from global generation, not from a descending chain condition for ideals. Fix a very ample divisor H_0. For each s, choose a Cartier divisor P_s so positive that

Asymptotic Nadel vanishing and Castelnuovo–Mumford regularity make all \mathcal J_s(t)\otimes\cO_X(P_s) globally generated [17, Corollary 1.5 and Theorem 1.8]. Their spaces of sections are nested subspaces of the fixed finite-dimensional vector space H^0(X,\cO_X(P_s)), and therefore eventually constant. Generation then makes the ideals themselves constant for all sufficiently small rational t>0.

For every divisorial u one has

The first bound follows from the log-resolution formula for a divisible level computing the asymptotic multiplier ideal. For the second, asymptotic subadditivity on the smooth variety X [17, Variants 2.4–2.5] gives, for sufficiently divisible m,

Taking orders, dividing by m, and taking the asymptotic infimum proves the bound. Combining stabilization, (3.4), (3.5), and (3.6), we obtain: for every fixed positive integer s, there is an index i(s) such that for every i\geq i(s) and every divisorial valuation u,

The index i(s) is independent of u. In particular,

Countability of finite limiting changes

We claim that there is a fixed countable subset \mathscr E\subseteq\R containing every finite limit of E_{i_a}(u_a), where i_a\to\infty, the u_a are integral-valued divisorial valuations, and A_X(u_a)\leq C for some constant C depending on the sequence.

Choose a countable algebraically closed field F\subset\C over which X and all the countably many ideals \mathcal J_s are defined. For every f\in F(X_F)^*, its zero and pole divisors on X have positive log canonical thresholds when nonzero. Their orders at u_a are therefore bounded by constants times A_X(u_a), so u_a(f) is bounded. Since these values are integral, diagonal extraction over the countable field makes them eventually constant for every f. The eventual values define an integral valuation u_\infty on F(X_F), trivial on F.

We next show that u_\infty is either trivial or divisorial. If it is nontrivial and its centre is not a divisor, blow up the closure of its centre and continue. At each stage we work near the generic point of the centre, where the ambient variety and that centre can be taken smooth. The next centre lies above this generic smooth neighbourhood, where the blowup is smooth. Taking a projective model and resolving outside it therefore does not change the valuation calculation. A centre of codimension at least two has an ideal of positive integral value. Blowing it up thus increases the value of the relative Jacobian over X_F by at least one. If N such blowups were possible, choose a smooth affine neighbourhood U=\Spec A of the last centre on which this relative Jacobian divisor is principal, with equation b. Then u_\infty(b)\geq N.

Choose finitely many F-algebra generators of A, including the inverses used to obtain this affine neighbourhood. Their u_a-values are eventually equal to their nonnegative u_\infty-values. Because each u_a is trivial on \C, every polynomial in these generators with complex coefficients has nonnegative value. Thus u_a has a centre on U_\C for all sufficiently large a. Its centre need not be the base change of the centre of u_\infty. The Jacobian identity holds throughout this chart and nevertheless gives

Taking N>C is impossible. The centre therefore eventually has codimension one. The valuation then dominates its discrete valuation ring, is zero on its units, and is a positive integral multiple of that DVR order.

There are only countably many projective models of F(X_F) defined over the countable field F, hence only countably many prime divisors on such models and their integral multiples. Adding the trivial valuation still gives a countable set of possibilities for u_\infty. For each fixed s, the preceding finite-chart argument applied to finitely many local generators of (\mathcal J_s)_F gives

If E_{i_a}(u_a)\to e, (3.7) therefore implies

One limiting valuation determines at most one such e: a second sequence may have a different bound C', but the two resulting limits differ by at most \max\{C,C'\}/s for every s. If u_\infty is trivial, all a_s are zero and e=0. This proves the claimed countability, including its independence from any subsequent choice of phases.

A common canonical index

We claim that there is a positive integer R such that

Otherwise, after passage to a subsequence, choose closed points x_i\in X_i whose local canonical indices r_i tend to infinity. If the indices were bounded, their least common multiple would give (3.10).

We recall the index-one cover at such a point. Choose a rational top form \eta, write K=\operatorname{div}(\eta), shrink about the point so that rK=\operatorname{div}(g), and normalize after adjoining z with z^r=g. Minimality of r and Kummer theory make this an extension of degree r. It is unramified in codimension one; ramification of discrepancies shows that it is klt. The form

has zero divisor and has primitive character under the cyclic cover group. The cover is Gorenstein: its canonical sheaf is generated by \theta, and klt singularities are Cohen–Macaulay.

There is exactly one point over the specified base point. Indeed, a finite group quotient has a transitive action on the points of each geometric fibre. If this orbit had more than one point, a nonconstant character of its cyclic permutation representation would give a regular semi-invariant a, of character z^j with 0<j<r, nonzero at every point of the fibre. One obtains a by lifting its values on the reduced fibre and projecting to the character space. Its invariant power a^r is nonzero at the base point, so a is a unit near the whole fibre. The function a/z^j is invariant, and its divisor downstairs is -jK. Thus jK is principal near the point, contradicting the minimality of r. The unique point is fixed by the cover group, as required in Theorem 3.1.

Now fix any \alpha\in\R/\Z. Because r_i\to\infty and the character of \theta_i is primitive, choose group elements \gamma_i so that the angles of the \theta_i-characters at \gamma_i tend to \alpha. Apply Theorem 3.1, and let u_i be the restrictions of the resulting w'_i to the common downstairs field. These restrictions are integral-valued and divisorial, and quasi-étale discrepancy compatibility gives

Choose a rational top form \eta_i^0 downstairs that generates the canonical sheaf at the centre of u_i on the smooth variety X. With pullback of forms understood, (2.1) gives

The ratio is a rational semi-invariant with the character of \theta_i. Consequently, after passage to the subsequence in Theorem 3.1, these nonnegative bounded changes have residues tending to \ell\alpha. Pass further so that they converge in \R. Their limit belongs to the fixed countable set \mathscr E, and hence \ell\alpha belongs to its image \overline{\mathscr E} in \R/\Z. But

is countable, whereas \alpha was arbitrary. The possible dependence of \ell on the subsequence does not change this contradiction. This proves (3.10).

Vanishing on late flips

For every fixed positive integer m, all sufficiently late flips, with negative contraction f_i:X_i\to Z_i, satisfy

To prove this, take a common smooth resolution p:U\to X_i, q:U\to X_{i+1} over Z_i, sufficiently high that the fractional parts to be rounded have simple normal crossing support, and put

The rational divisor (m-1)q^*K_{X_{i+1}} is nef over both X_i and Z_i: a curve contracted by p is also contracted over Z_i, and K_{X_{i+1}} is ample over Z_i. It is big relative to both bases since the relevant morphisms are birational; this includes m=1, whose divisor is zero and whose generic fibres are zero-dimensional. Relative Kawamata–Viehweg vanishing [28] therefore gives

We claim that p_*\mathcal L_m=\cO_{X_i}(mK_{X_i}), where the right side denotes the divisorial sheaf if mK_{X_i} is not Cartier. One inclusion is checked in codimension one, where the flip and the resolution are isomorphisms. For the reverse inclusion, let a local rational section h of \cO_{X_i}(mK_{X_i}) be given. The divisor

is effective, because \operatorname{div}_{X_i}(h)+mK_{X_i} is effective and rational Cartier. Before rounding, the difference between the divisor defining \mathcal L_m and mp^*K_{X_i} has coefficient, at any prime divisor of U,

For i\geq i(m), (3.8) implies

Adding this to the effective pullback of the section divisor and taking round-up leaves every integral coefficient nonnegative. Hence h belongs to p_*\mathcal L_m. The pushforward identity and Leray prove (3.12).

Fix R as in (3.10). For each late step choose a sufficiently positive very ample divisor J on Z_i so that the Cartier divisor

and D-K_{X_i} are ample. This is possible by relative anti-ampleness of K_{X_i}. Effective base point freeness, applied to the klt pair (X_i,0) and the nef Cartier divisor D with D-K_{X_i} nef and big, gives an integer N=N(n) such that L=ND is globally generated [18, Theorem 1.1 and Remark 1.2]. It is also ample. Only now choose a stage late enough for (3.12) to hold for the finite list

The integer N is independent of the step and of J, so this order of choices is legitimate. For each such step, projection formula, Leray and Serre vanishing on Z_i give, for all sufficiently large integers a,

Indeed the displayed divisor equals (R+jNR)K_{X_i}+(a-jN)f_i^*J.

Castelnuovo–Mumford regularity with respect to the ample globally generated L now makes \cO_{X_i}(RK_{X_i}+af_i^*J) globally generated. To use the usual very ample formulation, the morphism defined by L is finite onto its image, since L is ample; push the sheaf to projective space, apply regularity there using (3.14), and pull back the generating sections. A globally generated line bundle has nonnegative degree on every curve, contradicting its degree RK_{X_i}\cdot C<0 on a curve contracted by f_i. The hypothetical infinite program is impossible.

Pseudo-effectivity of the canonical class persists at every finite step: push forward the big systems of K_X+tH for arbitrarily small rational t>0, and take their limiting numerical classes. It rules out a final negative fibre contraction, since the restriction to a general positive-dimensional fibre would be both pseudo-effective and anti-ample. The terminal model Y therefore has nef canonical divisor. On a common smooth projective resolution p:U\to X, q:U\to Y, the MMP inequalities give

Taking P=q^*K_Y proves the last assertion.□

Proof. Replace Z by the Stein base of the morphism, which is normal and finite over its original image. The morphism f:X\to Z is then surjective with connected fibres. This replacement changes neither the contracted curves nor relative numerical equivalence, nefness or pseudo-effectivity. A very general fibre is smooth and projective by generic smoothness and has pseudo-effective canonical divisor. To see the latter assertion, take a countable sequence of relative effective approximations to [K_X] and restrict to fibres outside their countably many bad loci. Relative numerical equivalence restricts to numerical equivalence on such a fibre, and K_X restricts to its canonical divisor. By BDPP, these very general smooth fibres are non-uniruled [14, Theorem 0.2 and Corollary 0.3].

Compactify the projective morphism and resolve away from X to obtain a projective morphism

with \overline X smooth projective, \overline Z normal projective, and the original morphism unchanged over Z. If \dim Z=0, the variety X is already projective and Proposition 3.2 applies. We may therefore assume \dim Z>0.

Let r:\widetilde Z\to\overline Z be a smooth projective resolution. Choose a sufficiently high multiple H of a very ample divisor on \overline Z, and a general branch divisor B\in|2H|. The pulled-back linear systems on both \widetilde Z and \overline X are base point free. Bertini therefore makes r^*B and \overline f^*B smooth, nonempty reduced divisors. The corresponding double covers

are smooth integral projective varieties. Integrality follows because a nonempty reduced branch divisor cannot be the divisor of a square. The double-cover formula gives

For sufficiently positive H, the first canonical divisor is big. The rational map X^\sharp\dashrightarrow Z^\sharp has the same very general fibres as f. The base is non-uniruled because its canonical divisor is big, and those fibres are non-uniruled by the preceding paragraph. Hence X^\sharp is non-uniruled: a covering family of rational curves would either give a covering family on the base or consist of vertical curves covering very general fibres. BDPP now makes K_{X^\sharp} pseudo-effective.

Apply Proposition 3.2 to X^\sharp. On a projective birational model its canonical pullback is a sum of a nef rational divisor and an effective rational divisor. By (3.15), the same is true, after changing the nef divisor by a rational principal divisor if necessary, for the pullback of \pi^*(K_{\overline X}+\overline f^*H). Take a common equivariant resolution of this model and its conjugate under the covering involution. Writing g:T\to X^\sharp for the resulting morphism and \iota for the involution, average the decomposition and its conjugate. We obtain

where \overline P is nef and \overline N\geq0, both rational Cartier.

Let q:T\to Y=T/\langle\iota\rangle be the finite quotient. The normal projective variety Y is birational to \overline X. An invariant rational divisor descends as a rational Weil divisor by dividing its coefficient at an upstairs prime by the corresponding ramification index. If the invariant divisor is rational Cartier, its descended divisor is rational Cartier as well. Indeed, after clearing denominators, a Cartier divisor has one local equation on a semilocal neighbourhood of the finite orbit above a downstairs point. The product of its translates is an invariant equation whose divisor is the group order times the original divisor. It descends to a rational local equation for a multiple of the downstairs divisor. This is the finite norm argument, and applies at every point of Y.

Consequently \overline P=q^*P and \overline N=q^*N for rational Cartier divisors P,N on Y. Nefness of P follows by lifting any curve to T and dividing its intersection by the positive covering degree; effectivity of N is immediate from its coefficients. Finite pullback is injective on divisors, so (3.16) descends to a decomposition of the pullback of K_{\overline X}+\overline f^*H. Restrict to the inverse image of X. The additional term pulled from Z is numerically trivial over Z, and we obtain the required relative weak Zariski decomposition of K_X.□

Proof. Let k be such a field. Descend the given projective varieties, rational divisors and nef data, together with a log resolution carrying those data, to a countable algebraically closed subfield F\subset k. Embed F into \C. The initial properties hold over F and after extension to \C: generalized lc is checked on the chosen log resolution, and nefness is preserved under algebraically closed field extension by spreading and specializing curves. Apply the complex result. We will descend its finite MMP to F step by step; the chosen complex scaling divisor need not descend.

Suppose the current variety S and adjoint D are defined over F, and their complex extensions have been identified with the corresponding stage of the program. Consider its next complex contraction f_\C:S_\C\to T_\C. Pull back a very ample line bundle from T_\C and choose generating sections. This bundle and its sections spread on S\times_F B, for a nonempty integral finite-type F-scheme B. After shrinking B, they generate the bundle on every fibre: failure of generation has closed image under the projection, since S is projective. Specialize at an F-point and take the Stein contraction of the resulting map. This gives f:S\to T over F, with normal target and connected fibres.

After extension to \C, the specialized line bundle is numerically equivalent to the original one. Indeed, B is geometrically integral because F is algebraically closed. On its connected complex base change, the degree of the universal bundle restricted to any fixed projective curve on S_\C is constant. Hence the two contractions contract exactly the same curves.

We explain why this identifies their targets, including in the fibre-type case. Each morphism is constant on every connected projective fibre of the other: a positive-dimensional image of a fibre component contains a curve, and hyperplane sections of its projective preimage give a curve mapping onto it, contrary to equality of the contracted curves. The constants agree on intersecting components and hence on the whole connected fibre. The reduced joint image in the product of the two targets therefore has finite projections to both. After normalization, these projections are birational: the function field of a normal contraction target is algebraically closed in \C(S), so there is no nontrivial finite intermediate field. Both targets are normal, and finite birational morphisms to them are isomorphisms. Thus the descended contraction becomes the specified complex contraction. Stein contraction and normality are preserved by these algebraically closed extensions.

For a birational step, recover its positive model over T as

with b>0 sufficiently divisible. Over \C, the effective exceptional pullback difference (3.17) identifies these pushforward sheaves with the corresponding sheaves on the positive model, whose adjoint is relatively ample. Hence the relative Proj is exactly that model. This includes a divisorial step for which the positive model is T: the algebra is then the nonnegative section algebra of the Cartier multiple bD_T, whose relative Proj is T.

For clarity, both the sheaf algebra and its finite generation descend. On a normal variety over the algebraically closed field F, the sheaf of a Weil divisor is the rank-one reflexive extension of its invertible restriction on a regular open containing all codimension-one points. Field extension is flat with geometrically regular fibres in characteristic zero. The codimension-two complement retains its codimension, and the pullback sheaf retains reflexivity and the S_2 property. It is therefore the same reflexive extension as the sheaf of the extended divisor. Multiplication is also compatible, since it agrees on the regular open and its target is torsion-free. The canonical, boundary and nef-trace divisors defining D are all defined over F; the last assertion is checked by computing the trace on a common resolution.

Choose b clearing their rational coefficients and making the complex extension of bD Cartier. The divisorial sheaf \cO_S(bD) becomes invertible after the faithfully flat field extension and so is invertible already over F. The same b thus works downstairs. Proper flat base change identifies every graded pushforward in (3.19) and its multiplication after extension. On a finite affine cover of T_\C, finite generation of the complex algebra gives a common bound d_0 for the degrees of generators. Downstairs, form the subalgebra generated by its coherent pieces of degrees at most d_0. In each degree its inclusion in \mathcal R becomes surjective after faithful flat extension, so its cokernel is zero. Thus these finitely many coherent pieces generate \mathcal R, proving finite generation without presupposing it. Formation of relative Proj commutes with extension, so it supplies the next normal projective model and its birational map over F. This inductively descends the entire finite diagram, including its final fibre contraction when present.

Finally extend the diagram from F to k. Projectivity, normality, birationality, the absence of extraction on the positive side, connected contractions and relative ampleness persist. For a proper contraction, connectedness here is also expressed by the equality of the pushforward structure sheaf with the target structure sheaf, which is preserved by flat base change. Cartierness of the specified multiples has already descended from \C and is preserved over k.

We record the numerical facts needed for the endpoint. Every line bundle on an algebraically closed extension of a fixed projective F-variety is numerically equivalent to one from F: spread the bundle on a finite-type parameter scheme, specialize to an F-point, and use degree constancy on its geometrically connected base change, as above. Conversely, numerical triviality and nonnegativity of an F-divisor on relative curves persist under extension. Spread a vertical projective curve, together with the point of the target containing its image, in a flat projective family over a finite-type F-scheme. A specialization at an F-point is an effective vertical curve cycle with the same intersection degrees. The corresponding nonnegativity or vanishing over F therefore gives the same property for the original curve. The reverse implication is tested on base changes of F-curves. It follows that nefness and the relative numerical divisor spaces, hence relative Picard ranks, are unchanged. In the nef case the final adjoint remains nef; in the Mori case its negative remains relatively ample and the relative Picard number remains one.

Take common smooth resolutions of the descended finite steps, also dominating the original model carrying the nef divisor, and extend them to k. Their effective pullback differences remain effective, with the same positive coefficients along original contracted primes. The calculation (3.18) then gives weak increase of every generalized log discrepancy over k, with strict increase for each prime on the original X contracted by the composite. In particular the output is generalized lc. Its boundary is the pushforward of the original boundary and its nef part is the trace of the original nef b-divisor. The same effective pullback differences show that a nef endpoint makes the original adjoint pseudo-effective. Conversely, pseudo-effectivity of the original adjoint persists through the birational steps and rules out an anti-ample restriction to a general positive-dimensional fibre. Thus the sign of the original adjoint selects the stated alternative over k as well, completing Theorem 1.1.□

Proof. First take an eigenfunction f for the original grading torus, of positive weight. An equivariant log resolution gives an invariant divisorial valuation computing its threshold at the vertex. Normalize it to A(w)=1 and set c=w(f)>0. It is nonnegative on R. For a nearby rational normalized Reeb grading \zeta and a positive rational number s, define

on torus eigenvectors and take the minimum over characters on sums. This is a valuation with

Here is a direct verification of the discrepancy statement. On the product with a logarithmic line of coordinate z, take the Gauss extension of sw with w'(z)=\gamma>0 rational. Choose an integral one-parameter subgroup j and \gamma so that \zeta(\chi)=\gamma j(\chi), and apply the field automorphism h_\chi\mapsto h_\chi z^{j(\chi)}, fixing z. Its restriction has the displayed values u_s. It is still a Gauss extension: after clearing denominators, coefficients that tie at a power of z do not cancel across distinct characters, because the w-filtration is torus invariant. The transformation of \theta\wedge d\log z adds the canonical character weight 1 to discrepancy. The original Gauss extension has discrepancy s. Both it and the transformed restriction are divisorial up to rational scaling: the value group is discrete and nonzero, and restriction from the product leaves residue transcendence degree at least n-1. The logarithmic Gauss extension has the same discrepancy as its restriction. These observations prove (4.6).

Put d_\zeta=\zeta(f). Multiplication by f shifts the u_s-filtration exactly, while u_s\geq\zeta on R. Counting modulo consecutive powers of f therefore gives

Indeed, at cutoff p the length is at most the sum, over k\geq0, of the dimensions in R/(f) up to \zeta-weight p-k(d_\zeta+sc). The cumulative Hilbert growth of R/(f) has leading coefficient d_\zeta\vol(\zeta)/(n-1)!, by the nonzerodivisor identity for the rational graded Hilbert series, after integral rescaling. Summing this polynomial growth proves (4.7); local lengths at the vertex equal these affine counts.

Use normalized-volume minimization on C_0 and let \zeta approach its minimizing Reeb valuation. Multiplicative comparison of weights gives continuity of volume. Writing d_*=v_*(f), we obtain

Letting s decrease to zero yields c\leq nd_*. A homogeneous h_I for the rational regrading may have several components for the original torus. A one-parameter torus degeneration selects a nonzero extreme component. Semicontinuity of the threshold in the trivial family, equivalently inversion of adjunction, bounds the threshold of the general translate below by that of this component. Its original weight is at most 2I/a. This proves (4.3).

For a nonunit g, the function t^{-v_{\mathrm{int}}(g)}g is regular near the special point of the Rees family and restricts to its graded initial. Inversion of adjunction for a normal Cartier divisor in a Gorenstein variety [26] transfers the threshold bound to the general pointed germ. Its different is restriction here, since the total space is smooth along the smooth locus of the central fibre. More explicitly, the pair obtained by adding the central divisor is lc near the special point; away from t=0 the family of divisors is a product, and the pointed section gives the desired bound at o. This is exactly (4.4) for divisorial valuations. Approximation on a log-smooth model gives it for real quasi-monomial valuations. Units have value zero and cause no exception.

The monomial generators of valuation ideals proved above give (4.5): each generating monomial of v-weight at least v(g) has w-weight at least s(w)v(g). Finally a nonunit at the vertex has threshold at most one, so (4.3) implies the lower bound I/a\geq1/(2n).□

Proof. Spread P to a horizontal divisor valuation P_0 on the normalized graph over the rational base. A finite algebraic extension of the generic base used to label its component is unramified at this horizontal divisor. By separability on the divisor, the order of the relative differential form is its absolute order before division by the base differentials. Thus we can compute using P_0.

When c_P=0, its generic centre lies off D. The rational map is regular there, so its normalized graph is isomorphic to that open of \widehat E, where \eta has zero Weil divisor. This gives discrepancy one. Suppose c_P>0 and let S be the centre of P_0 on \widehat E, of codimension r\geq1. Among the residues of the \phi_j there are r-1 algebraically independent elements over k(S). To see this, the residue field of P_0 has transcendence degree n-2, and is algebraic over the field generated by k(S) and the base coordinates: normalization of the graph is finite over the graph. All these base coordinates are units at the horizontal divisor, so a transcendence basis can be chosen from their residues. The difference of transcendence degrees is r-1.

Extend P_0 to k(\widehat E)(u) by the Gauss valuation \nu with \nu(u)=c_P/a, and also restrict \nu to the original cone field. It is nonnegative on homogeneous regular elements: the divisor of a degree-I section divided by u^I is bounded below by -IH_u. Thus \nu has a centre on the affine cone cover.

The field of this affine centre has transcendence degree n-r and is algebraically related to k(S)(\tau) over a common subfield of that transcendence degree. Here \tau is the residue of a positive-degree section that is a unit for \nu. More explicitly, choose a very ample Cartier multiple and a section nonvanishing at S; its residue is transcendental over the residue field of P_0 by the Gauss construction. Ratios of other sections in that multiple recover the dimensions of S. For every other homogeneous unit, taking powers and ratios with the chosen section makes its residue algebraic over k(S)(\tau). Homogeneous decomposition proves the same assertion for residues of units in the entire affine ring. The finite map to \Spec R preserves the dimension of this centre.

Choose n-r regular functions on \Spec R with independent residues on that centre. The selected r-1 residues of the \phi_j, or their positive powers, remain independent over its centre field, because \tau is Gauss-transcendental over the residue field of P_0. Together with f_1, use the corresponding r-1 functions among the f_j. After taking powers, their degree-zero ratios with f_1 give the chosen powers of the \phi_j.

These n cone functions have a logarithmic differential wedge of discrepancy zero at \nu. In fact monomial combinations with nonzero determinant give one function of positive order and n-1 units with independent residues. In characteristic zero their logarithmic wedge has a simple pole along the divisorial valuation. Hence their ordinary differential wedge has discrepancy equal to the sum of their values, which is at most \sum_{j\leq m}e_jc_P. Dividing that wedge by \theta gives a regular coefficient on the cone: this holds at every codimension-one point, and normality extends it. On the other hand, (4.11) and the logarithmic Gauss formula give

Nonnegativity of the coefficient’s value proves (5.5). If m=1, necessarily r=1 in this calculation; the list of \phi_j is empty and the same proof applies.□

Proof. The form \eta_F provides a section. Its poles are simple and lie over D_F, so the positive ceiling supplies at least one copy of each pole. Suppose a second section has nonconstant ratio with \eta_F. After extending scalars to the algebraic closure of the rational base field, regard F as a component of a general smooth complete intersection of the free pencils on W. Extend the section by zero on the other components. Let M_j be the pole divisor on W of \phi_j; it is a member of the corresponding free pencil.

Successive adjunction lifts this section to

To give the restriction step explicitly, on an intermediate smooth component Y write A=bTD_Y, let C be the next general member of its free pencil, and let R_Y be the sum of the remaining pencil divisors. Adjunction gives the exact sequence

The first term has vanishing H^1 by Kawamata–Viehweg vanishing [19, Section 3.1]. Indeed, B_Y=\lceil A\rceil-A has SNC support and coefficients in [0,1), and the difference of the displayed integral divisor and K_Y+B_Y is A+R_Y, which is big and nef. Bigness on each intermediate component follows, as for F, from its image on the original domain of the graph. All supports were resolved before taking the cuts, and ceilings commute with these cuts as explained above. In the finite base chart, adjunction by the equations \phi_j-\lambda_j identifies the lifted ratio with the required ratio of relative forms. Denote the lift, written as a rational multiple of \eta, by H'\eta.

On the base-changed \widehat E, the poles of H' are bounded by

Indeed, at a prime with coefficient c>0 in D, the case m=1 of Lemma 5.3 gives \ord\eta\leq(N-1)c-1. The ceiling excess is at most one, and the pole divisor of each \phi_j is at most e_jD. Adding these inequalities proves (5.7). At a prime with c=0 the form has order zero and these divisors have no pole.

The selected root field is cyclic Galois over the original cone field, of degree dividing aN. Split H' into its deck-character parts, with the enlarged scalar field fixed. Each part still obeys (5.7), because D is invariant; the deck group is not required to preserve the selected geometric fibre. For a part of given character choose an integer I in the cancelling congruence class with

Multiplication by u^I makes it an invariant homogeneous regular function on the cone cover. Regularity follows in codimension one from the pole inequality and the divisor H_u; the vertex has codimension at least two, so normality completes the check. This invariant is therefore in the scalar extension of R_I.

The preceding degrees are negligible compared with T, and a\geq1. Since bT\leq T/10, one index in the sequence sufficiently far along the jump makes I/a<T for every rational b in the asserted range. Every ground-field basis element h\in R_I is then algebraic over k(f_1,\ldots,f_m). Its quotient h/u^I is algebraic over k(\phi_2,\ldots,\phi_m). To check the latter assertion, after adjoining u the original algebraic relation is a relation over k(\phi_2,\ldots,\phi_m)(u); but h/u^I belongs to k(\widehat E) and u is transcendental over that field. Expanding a relation in powers of u gives a nonzero relation over the base field alone. Such a rational function is constant on each connected geometric generic component. This remains true for scalar combinations of the basis elements. Thus every deck-character part of H' is constant on F, contradicting the chosen nonconstant ratio.□

Proof. The restriction of v to k_0(f_1^0,\ldots,f_n^0) is the weighted Gauss valuation. The weight-zero Laurent monomials for a basis of the kernel of \Z^n\to\Q, \alpha\mapsto\sum\alpha_jd_j, have algebraically independent residues. Refine v by a full-rank monomial valuation on these residues, and extend this refinement to the finite extension of residue fields upstairs. We obtain a composite valuation \tau with residue field k_0. Using \tau(f_j^0) as coordinates, its value group is a lattice \Gamma\subset\Q^n containing \Z^n with finite index, and the original real value is the pairing with (d_1,\ldots,d_n).

Suppose centred regular functions l_1,\ldots,l_n have independent \tau-values generating a sublattice \Lambda. Every class of \Gamma/\Lambda has a representative which is the value of a regular function: the regular value semigroup generates \Gamma, and its image in a finite group is a subgroup. Fix one such representative per class. Multiplying by the monomials l^\nu, \nu\in\Z_{\ge0}^n, gives distinct \tau-values. Those of v-value at most T are independent modulo (v>T). Comparing the leading coefficients as T\to\infty gives

The fixed representatives contribute only bounded shifts in this asymptotic comparison; no uniform bound for those shifts is needed.

Applying (6.4) to f^0 gives [\Gamma:\Z^n]\le C. If the jth coordinate \tau(g)_j of a regular nonunit g is nonzero, replace f_j^0 by g. The determinant of the new lattice basis is |\tau(g)_j|, so

Consequently

The inequality is automatic when that coordinate is zero. Units have \tau-value zero.

If v(g)\le T<d_{m+1}, the initial of g is algebraic over the graded fraction field generated by the initials of f_1^0,\ldots,f_m^0. Take a homogeneous polynomial relation. On refining its values by \tau, at least two lowest terms tie, and they involve distinct powers of the initial of g: coefficients belong to the graded field generated by an independent tuple, where their nonzero initials cannot cancel. Thus \tau(g) lies in the rational span of the first m coordinate vectors. The number of lattice points in this subspace satisfying (6.5) and v(g)\le T is at most C\prod_{j\le m}(1+T/d_j)\le C'\prod_{j\le m}(T/d_j). Here one can place \Gamma in \frac1e\Z^n with e=[\Gamma:\Z^n]\le C. Finally \tau has one-dimensional leaves. The bounded box has only finitely many lattice points, so successively taking initials in the finite jet quotient bounds its dimension by this count. This proves (6.3).□

Proof. For (i), make a unimodular monomial change in the tuple so that the first r_0 weights are rationally independent and the remaining weights are zero. Their residues are algebraically independent. Resolve the supports of the first functions and the rational maps to \mathbb P^1 defined by the others on a proper smooth model. At the centre, the latter functions are units with independent restrictions, so the codimension is at most r_0. At least r_0 boundary components are needed to express the independent nonzero weights. Thus the centre is a generic stratum of codimension r_0, its boundary parameter weights are independent, and the valuation is the corresponding monomial valuation. Indeed different parameter exponents cannot tie, and the valuation of sufficiently high powers of the maximal ideal tends to infinity. The vertical exponent matrix of the first functions has nonzero determinant, and the horizontal residue differentials of the remaining functions are independent. Their log wedge is therefore a unit log volume at the stratum, proving (6.6). Applying the same calculation upstairs gives the finite-extension assertion, consistently with the ramification formula for form-discrepancies.

For (ii), complete f to a full homogeneous-independent tuple t. This is possible because the graded field of a quasi-monomial valuation has transcendence degree n, by the Abhyankar equality. Change the prescribed weights of f, keep the other tuple weights fixed, and use Gauss valuations on k_0(t). The field L is finite over this rational field. For an algebraic element, its possible extension values are the negatives of Newton-polygon slopes of its minimal polynomial, with multiplicities. These form a finite multiset depending continuously on the coefficient values. One may see this without completeness by extending to an algebraic closure, factoring the polynomial, and using multiplicativity of Newton polygons.

Several specified values must be tracked by one extension. For a finite list of elements, choose integer powers so that the value of their product separates all distinct numerical tuples in the finite Cartesian product of their candidate value sets at the starting point. Choose a root of the product’s minimal polynomial with value tending to the prescribed product value. The resulting embedding of the product field into the algebraic closure extends to L. Under this single embedding every individual value belongs to its candidate set. Separation forces all those values to converge to the prescribed ones. Include in the list the coefficient of \eta in the full log frame, and the functions h_i. Formula (6.6) gives discrepancy convergence, and positivity of the h_i preserves the centre. This argument asserts no uniform size of the neighbourhood.

For (iii), at a generic stratum write numerator and denominator in the completed regular local ring with a coefficient field and the boundary parameters. The minimum of the occurring exponent weights is determined by finitely many exponents, by the Noetherian property of monomial ideals. This proves rational piecewise linearity. Setting some weights to zero gives the same evaluation at the containing stratum. More explicitly, monomial cutoff ideals in the remaining parameters are primary for the prime generated by those parameters, and localizing to that stratum and contracting leaves them unchanged. The same assertion is visible in the power-series completion. This proves continuity along every face.□

Proof. The discrepancy formula on a log-smooth model separates into the boundary-linear contribution and A_{Q,D}(w). Retraction retains the former. For the equality statement, subdivide the orthant fan near the centre by rational hyperplanes respecting the smallest rational subspace containing the boundary-weight vector, and then take a regular toric refinement. Such subdivisions are realized locally by toroidal modifications: boundary equations extend to smooth, or etale, coordinates, so the construction pulls back from coordinate-axis charts. The reduced pair remains crepant.

The lifted centre lies on the open toric stratum specified by the relative-interior cone of the weight vector. Weight-zero chart monomials are units there. The positive normal weights are rationally independent, since their cone lies in the smallest rational subspace just chosen. If the centre is not the generic point of that stratum, choose, near its generic point, a smooth divisor through the centre transverse to the boundary. Adding it to the boundary subtracts a strictly positive value from the discrepancy, while the resulting smooth SNC pair is lc. Thus A_{Q,D}(w)>0. If the centre is generic, independent positive parameter weights force the monomial valuation, as in Lemma 6.2.

The monomial valuations on the refined charts are precisely those on the subdivided original complex. Indeed the new normal parameters are toric monomials, and the weight-zero ratios of old boundary monomials restrict to characters of the torus stratum, with only the monomial lattice relations at its generic point. The monomial calculation therefore pulls back unchanged from the orthant charts. This identifies the equality case on the original complex.□

Proof. Resolve the supports of h_i,f_j,\eta on a smooth proper model (Q,D). Retraction preserves the values of these functions and s. It therefore remains in the prescribed fibre, and preserves the Gauss condition because b is independent. By Lemma 6.3, an attained minimizer must lie on the complex.

To construct detectors, take local rational parameter equations for the boundary components at the generic stratum of each cone. Each parameter is algebraic over L^G. Take a polynomial equation for it and include invariant regular numerators and denominators for all its nonzero coefficients. Such presentations exist: an invariant rational function acquires an invariant regular denominator by multiplying a denominator by its other group translates. Once the detector values are fixed, Newton root values allow only finitely many values for each parameter. Since there are finitely many cones, simultaneous detector fibres on the complex are finite.

Subdivide so that the evaluations in question and s are linear. On the closed cones cut out by w(h_i)\ge0, every nonzero point has A_\eta>0. To check this, first use rational rays, which are divisorial and have affine centres because the h_i include affine generators; klt gives positivity. Linearity on the finitely many rational cones then gives positivity and properness on each cone. For s>0, (6.1) gives

Thus a bounded sublevel of \Phi has bounded A_\eta. A limit cannot have s=0: boundedness of \Phi would force A_\eta=0, hence the zero valuation, contrary to w(f)=b\ne0. The sublevel is therefore compact, and both the minimum and the secondary maximum are attained.

On each cut polyhedron the minimum locus is the compact face A_\eta-\Phi_{\min}s=0, disjoint from s=0. Maximizing the linear detector sum on that face allows a vertex of the original cut polyhedron. Its coordinates solve rational linear equations with right side linear in b. Hence they are rational-linear in b. All function values are integral combinations of finitely many parameter weights at its stratum; this proves the common denominator assertion. Since the values of the f_j are the independent b_j, the rational rank is exactly r. The active generator identity follows by independence of b.

There are only finitely many vertex formulas and polyhedral feasibility conditions, including s>0. Comparisons of their ratios and then detector sums are semialgebraic. Restriction to a segment therefore has a finite interval partition. On a valid branch the denominator is positive, so its ratio is rational-smooth. For the last assertion, take a convergent subsequence in the fixed finite complex. The compactness argument keeps s>0. For every fixed polynomial P in the tuple, each independent-weight valuation satisfies

Both sides are continuous on the complex by Lemma 6.2. The single limit valuation consequently satisfies this equality for every polynomial, even when monomial weights tie. No further subsequence depending on P is needed. It is therefore Gauss, and admits the approximate continuation of Lemma 6.2.□

Proof. Both assertions are homogeneous in z, so we may take z primitive. Consider the lexicographic composite valuation (\widetilde w,z), whose second value on a\in\widetilde L^* is z(a'). Adapt the residual tuple so that \bar y_1 is a parameter for z and the other \bar y_j are units with independent residues at its generic centre. The tuple x,y is Gauss for its two rows of lexicographic weights. On its rational function field replace those rows by the first row plus t>0 times the second row.

We require extensions which track exactly, and simultaneously, the projected lexicographic values of finitely many elements. For each test element, take its polynomial over the full tuple field. For sufficiently small t, each coefficient value is the projection of its lexicographic value. Pairwise equalities of term values give a finite set of lexicographic candidates for the element’s value. Choose integer powers of the test elements so that their product value separates all distinct tuples in the Cartesian product of these finite candidate sets; include all candidates, whether or not they are lowest values. Its actual lexicographic value is a Newton root value of its minimal polynomial. The finitely many comparisons determining the Newton polygon are preserved after projection for all sufficiently small t. Hence the projected product value is the value of a root over an algebraic closure of the projected valued tuple field. This uses the Newton-polygon fact that a value where the minimum of the term values is attained at least twice is a root value for a polynomial with nonzero constant and leading coefficients, as follows by factoring over a valued algebraic closure. The embedding of the product field sending it to this root extends to the whole finite extension \widetilde L. The values of all test elements under that one embedding belong to their projected candidate sets. Distinct separating product values remain distinct for small t, so this forces the entire desired tuple of values.

Include the coefficient of \eta in the adapted log frame, all h_i, and every detector. The parameter y_1 already has prescribed weight t on the tuple field; it can also be included in the list. The resulting extensions, and their restrictions w_t to L, are quasi-monomial. Indeed full graded transcendence degree is preserved by finite extension and restriction, and the full-tuple argument of Lemma 6.2 applies. Discrepancies of pulled-back forms agree with those on restriction. Exact tracking therefore gives, for all sufficiently small t>0,

In the second equality inactive generators remain inactive for small t. In the first, the coefficient in the adapted residual log frame has order A_\rho(z). The h_i retain positive values, so w_t stays centred; the f_j=x_j^{N_0} retain their weights b_j. Minimality of \phi now gives (7.6).

Suppose equality holds and z\ge0 on A_0. Equations (7.7) imply \Phi(w_t)=\phi. Strict retraction places every w_t on the original cone complex. All detector values are nondecreasing, since their normalized initials lie in A_0. Their sum was maximal, so every detector value is constant. The joint detector fibre is finite. Each of its valuations on L has only finitely many extensions to the fixed finite field \widetilde L. This contradicts the actual variation \widetilde w_t(y_1)=t. Thus equality cannot hold over the affine part. When q=0 there are no divisorial z, and both assertions are vacuous.□

Proof of Proposition 7.1. Set Q_p=H\prod_\nu t_\nu^{j_\nu}f^k. Choose k in the indicated box so that m_{Q_p}+e\in p(\frac1{N_0}\Z^r). This is possible because p\nmid N_0. Define

Ordinary Cartier sends regular top forms to regular top forms on the smooth locus. There \eta is a canonical generator, so R is regular. Normality extends it across the complement of codimension at least two. Once nonvanishing is known, semilinearity and compatibility with the group action give (7.2).

Compute the value upstairs, where Cartier commutes with separable pullback, as follows also from its field p-basis formula. The tuple x,y is an absolute p-basis over the perfect constants. Indeed its monomials with exponents in [0,p) are independent over pth powers already in the graded field: use the basis of the value lattice for x and the separating residual p-basis for \bar y. There are p^n of them, so they form a field basis. In the log frame of (7.4), write the coefficient of Q_p\eta as

Lowest terms cannot cancel by that graded independence. Since m_c=m_{Q_p}+e lies in p(\frac1{N_0}\Z^r), lowest terms have I=0. Normalize by x^{-N_0m_c}, a pth power. The residue times d\log\bar y is

Its residual Cartier image is nonzero by (7.17) and (7.19). Thus the term with I=J=0 occurs at the lowest value, and m_{c_{00}}=m_c/p. Log-frame Cartier extracts precisely c_{00} times that frame. Division by \eta gives m_R=(m_{Q_p}+e)/p-e, proving (7.1) and nonvanishing. The selected H depends on p but is unchanged for every power vector: multiplication by any product of the nonzero constants t_\nu' cannot kill its residual Cartier image. Thus no infinite list of products needs to be spread.

For the final assertion, fix a cutoff T in characteristic zero. The ideal I_T=(v>T) is primary to the maximal ideal, since sufficiently high powers of its finite set of generators have v-value greater than T. Choose finite generators of I_T, identities expressing containment of a high maximal-ideal power, and finite identities for G-stability. Spread these, keeping any local denominators invertible. After shrinking, the finite quotient has constant length \mathcal H(T). Track the chosen generators in the monomial chart just described. By (6.1), each has characteristic-zero value strictly greater than s(w_{\rm orig})T, and its value is unchanged on the reduction. Nonnegativity on the reduced local ring therefore gives

There is no assertion that the whole valuation v is spread. If reduced eigenfunctions with distinct (value, character) pairs of weights at most this cutoff had a dependence modulo I_{T,p}, split it into characters using stability and p\nmid|G|. Within a character the distinct lowest value cannot cancel, contradicting (7.20). Their number is therefore at most the unchanged quotient length.□

Proof. For (8.8), take any finite list of functions representing distinct input classes. Use them as the tests in Proposition 7.1. For each test take its power one and all other powers zero, using the same H for these outputs. For arbitrarily large p, (8.5) holds. If two output classes were equal, subtracting (7.1) and (7.2) would show that the corresponding input classes are equal: the k terms belong to \Lambda_f, and multiplying an element of \Lambda_f by p remains in \Lambda_f. No division by p in a character group is used. Thus these outputs give disjoint boxes (8.7), and the size of every finite list is at most C_3.

If w(t)=0, independence of b gives m_t=0. Otherwise apply the same proposition to the powers of this one test with 0\le j/p\le d/w(t). Its output’s ith exponent coordinate is a common translate plus (j/p)m_{t,i} and an error in [0,1). The common o(1) error from (7.18) is independent of j. If |m_{t,i}|d/w(t) exceeded a sufficiently large fixed multiple of d/d_i, choose more than C_3 evenly spaced values of j/p in this interval. For large p they can be rounded to the 1/p grid with arbitrarily small additional error. Since d/d_i\ge1, their output coordinates can be made more than d/d_i apart. The corresponding boxes are disjoint, contradicting their uniform bound. This proves (8.9).□

Proof. Suppose all regular eigenfunctions of w-weight at most L_*d have algebraic initial. At any fixed pair (m_t,\chi_t) the initials span at most one dimension: after adjoining x and normalizing, (7.19) puts them in k_0. By (8.8) there are at most C_3 classes, and within one class the shift in \Lambda_f is determined by its integral exponent vector. By (8.9), the number of possible pairs is at most

Now use the original tuple, not the changing optimized one. For a fixed large D\ge1, consider the span of

Its dimension is at least D^m\mathcal D_m. Every nonzero combination has v-value at most mDd, because the original initials are independent. By (6.1) and (8.2), its w-value is at most C'Dd. Decompose this finite-dimensional space into character subspaces and filter each separately by w. This does not require the w-filtration itself to be group-invariant. A valuation filtration on a finite-dimensional vector space has a basis adapted to its finitely many occurring values, so the dimensions of its initial spaces sum to its dimension. If all these eigenfunction initials were algebraic, (8.10) with L_*=C'D would bound that dimension by C(1+C'D)^r\mathcal D_r. Since r<m and \mathcal D_m\ge\mathcal D_r, a sufficiently large fixed D makes this impossible. This supplies t with the asserted uniform bound.

The function is a nonunit, because the initial of a unit centred at the closed point is a nonzero scalar. Its transcendental initial enlarges the Gauss tuple. The degree d_{r+1} belongs to the current tier, so d/d_{r+1} is uniformly bounded. Finally apply Lemma 6.2, retaining the h_i, to move to independent weights as close as desired. Discrepancy and s>0 are continuous in that continuation, so one can reserve at most one unit of increase in \Phi.□

Proof. Apply Proposition 7.1 with no t_\nu. Its output has

Fix a sufficiently large integer D and consider, in the reduction, all boxes

Their weights are at most sKDd after increasing the uniform K. If |m_{R,i}|>Dd/d_i for some i, the boxes have disjoint exponent coordinates in that direction. Within each box the independence of b gives distinct values. There would be at least D^{m+1}\mathcal D_m distinct values. The same cutoff-inclusion argument applies to the regular eigenfunction products R^jf^\alpha, and bounds their number by C_WK^mD^m\mathcal D_m. Choose D>C_WK^m, fix this cutoff and all data, and then let p be arbitrarily large. It follows that |(\phi g_s-e)_i|\le(D+1)d/d_i, with harmless adjustment of the constant for the o(1) term. Divide by the lower bound for s in (8.2) to obtain (8.11).

On a fixed branch, A_\eta=e\cdot b and s=g_s\cdot b are rational-linear formulas. The derivative of their ratio is ((e-\phi g_s)/s)\cdot\dot b. At independent parameters the coefficient vectors are exactly those of the selected vertex, so (8.12) follows.□

Proof. A constant segment requires no change. Otherwise independent parameters are dense along the segment: for each nonzero rational vector a rational dependence is an affine equation in the segment parameter, not identically zero because the starting weights are independent. Apply the finite interval partition of Proposition 6.4. Initially approximate continuation supplies feasible independent assignments with limiting optimal ratio at most M.

On each open interval where a selected branch is feasible, integrate (8.12) as long as the ratio stays within the reserved bound M+1. The estimate holds at the dense independent parameters and hence everywhere on the interval by smoothness of the rational branch formula. At a terminal parameter of such a bounded interval, selected vertices at independent parameters have a compact convergent subsequence on the fixed complex: s\le2, A_\eta is bounded, discrepancy is proper on the relevant cones, and (8.2) keeps s positive. Its limit is Gauss on the tuple by Proposition 6.4, including all polynomial identities at these possibly dependent weights. Lemma 6.2 continues this valuation to nearby independent assignments to the right with limiting ratio no larger than the endpoint value. Thus there is neither loss of feasibility nor an upward jump at a branch endpoint.

Proceed in order through the finitely many intervals of the partition. On an interval, stop at a putative first excess of the reserved budget; until that time all constants are valid. With the segment parameterized by 0\le t\le1, integration and (8.15) propagate the bound

The limsup argument propagates it across every endpoint. Since C_6C_7/S<1, a first excess of M+1 is impossible. The polyhedral selection may be relaxed at dependent parameters inside intervals; this causes no difficulty, because the selected branch is actual at the dense independent parameters where the derivative is tested, and compact limits provide actual Gauss extensions at endpoints. At the final endpoint take the compact limit itself.□

Completion of the proof of Theorem 3.1. It remains to pass from the congruences on the completed tuple to every rational semi-invariant, with one fixed additional integer. Choose a uniform positive integer D_{\rm fin} so large that

For any regular nonzero eigenfunction h, take T\ge d_n sufficiently large and consider the indexed functions

The number of indices is at least

Choose T still larger so that D_{\rm fin}w(h)\le sT/2. Every indexed function then has w-value at most sT. There is a nontrivial linear dependence modulo (v>T), whether or not some of the indexed functions already coincide. Since v is invariant, the ideal is stable and the dependence can be chosen within a single character. Its sum either vanishes or has w-value greater than sT, by (6.1). Hence the lowest terms must cancel in the w-graded ring.

This cancellation involves at least two distinct exponents j,j'. Otherwise, after factoring the nonzero initial of h^j, a nonzero polynomial in the initials of f would vanish. This contradicts their Gauss property, even though the target weights have rational relations. Choose two lowest terms with distinct powers. Equality of their weights and equality of their characters, together with (8.16), give

Set

Then for every regular nonzero eigenfunction,

The cutoff T may depend on h and on the germ; the integers D_{\rm fin},\ell,S do not. In particular the same \ell works for all these functions at once.

For a rational semi-invariant h, choose a regular denominator b and replace it by the product of its group translates. This is a nonzero invariant regular denominator b_G, and hb_G is regular and has the same character as h. Subtracting the two already established congruences proves the assertion for h. On invariant rational functions the resulting value is integral. Finally w' is centred and divisorial-scaled, and its discrepancy is bounded by \ell S C uniformly on the retained subsequence. This is exactly Theorem 3.1.

The only input whose proof has been deferred is Proposition 5.2, used in Proposition 5.1. The remaining sections prove that proposition and thereby complete the dependency chain.□

Proof of Proposition 5.2, assuming Theorem 9.1. Write A=A_{\eta_F} and l_w=w(L). The dimension of F and the clipping constant in Proposition 5.2 are fixed along the sequence. We begin with a rational t>0, which will be chosen only after obtaining constants independent of t. Use the canonical divisor K_F=\operatorname{div}(\eta_F) and put

The boundary \Delta has coefficients in [0,1) and SNC support, so (F,\Delta) is klt. Since \eta_F is an lc volume form, every coefficient of K_F is an integer at least -1. A coefficient -1 gives A(P)=0 and hence l_P>0. Thus E is an effective integral divisor.

Choose a sufficiently divisible q and a general H_q\in|qtL|, and set H=H_q/q. Semiampleness permits H_q to be chosen smooth and transverse to the fixed SNC support. Taking q\ge2, the pair (F,\Delta+2H) is sub-lc. Consequently

Here and below discrepancies carrying a pair subscript are ordinary pair discrepancies. Define forms \sigma,\psi by

The first form is \eta_F, of index one; the second exists because tL-H\sim_{\Q}0. From

and (9.2), we obtain

Strict positivity uses the hypothesis l_w>0 at every nonzero divisorial valuation with A(w)=0.

The pair (F,\Delta+H) is klt, its boundary is big, and its adjoint is rationally linearly equivalent to E\ge0. The good minimal model theorem of [11] gives a \Q-factorial good minimal model Y. Let h:Y\to S be its semiample contraction, with connected fibres. Subscripts denote birational transforms. The rational principal divisor tL-H is the same on every model, so the ordinary MMP discrepancy inequality gives, on a common resolution,

Here N is an effective ample \Q-Cartier divisor when \dim S>0. To obtain equality of divisors, take a globally generated multiple of the effective semiample divisor E_Y. Its defining section descends to the ample system on S because h_*\cO_Y=\cO_S, and its zero divisor defines that multiple of N. If S is a point, the effective divisor E_Y is rationally linearly trivial and is zero; we then put N=0. The effective pair (Y,\Delta_Y+H_Y) is klt, and

We claim that every integral-valued divisorial valuation satisfies

If the conclusion fails, the nonnegative integer A_\sigma(w) is zero. Let D be the reduced SNC union of the supports of K_F and L. If k_P is the coefficient of K_F at a component of D, then

All summands are nonnegative. Thus every component seen by w has k_P=-1 and l_P>0. Since w(P) is a positive integer,

The support of E is contained in D, so w(E)=0, contradicting (9.5). This proves (9.7); its unit threshold is exactly where the index-one hypothesis on \eta_F is used.

9.2. A uniformly positive base polarization

Suppose \dim S>0, and put P=\C(Y) and Q=\C(S). Connectedness makes P/Q regular. On the generic fibre of h, the boundaries of \sigma and \psi are respectively

restricted to that fibre, because h^*N restricts to zero. They are effective. The second gives a klt log Calabi–Yau pair with big boundary: H_Y is big by birational pushforward and its restriction to the generic fibre is big. A klt log Calabi–Yau pair with big boundary is of Fano type. Explicitly, write its boundary as a rational linear equivalence B\sim_{\Q}A_0+D_0 with A_0 ample and D_0\ge0. For sufficiently small e>0, (1-e)B+eD_0 is effective klt and its negative adjoint is rationally linearly equivalent to eA_0. If the generic fibre has dimension zero, regularity gives P=Q.

Apply Theorem 9.1(1) with reference index one, comparison constant 2, and, for example, \lambda=1/4. By (9.4) and (9.7), an integral-valued valuation lying over a positive multiple of a valuation u with u(N)>0 has A_\psi\ge1/2. Thus there is \delta>0, depending only on the fixed dimension and independent of t, such that

In relative dimension zero this follows directly from the barrier; decreasing \delta accommodates that case as well.

Apply the lc-trivial canonical bundle formula with b-semiample moduli part, [5, Theorems 1.5 and 6.28], to h:(Y,\Delta_Y+H_Y)\to S. The morphism is projective and connected; the boundary is effective and klt, and

The discrepancy rank condition is one: on a generic resolution, the rounded discrepancy divisor has no negative coefficients and its positive coefficients occur only on exceptional divisors. Its section space is therefore the constants on the connected generic fibre. These verify generic effectivity, generic sub-log-canonicity, rank one, and the rational adjoint pullback required by the cited theorem.

Its generalized base discrepancy is

Indeed, after putting u on a smooth base model, the lc threshold over its DVR is computed by resolving the fibre divisor together with the boundary. It is the minimum of the finitely many discrepancy-to-multiplicity ratios of vertical components; the horizontal directions are already klt. Thus the infimum in (9.9) is attained and is positive. On the original S the discriminant is effective: over the generic point of a prime of S there is a prime of Y of positive integral fibre multiplicity, and its ordinary discrepancy is at most one.

On a high smooth model where the moduli divisor is semiample and the crepant discriminant has SNC support, replace the moduli divisor by H_0/q_0, where H_0 is a general member of its q_0th multiple and q_0\ge2 is sufficiently divisible. The subpair obtained by adding H_0 with coefficient one is sub-lc, so u(H_0)\le\beta(u) for all u. After pushforward this gives an effective ordinary klt boundary B_S with

Both the discriminant trace and the pushed-down general member are effective. On every model the replacement changes the adjoint by the same rational principal divisor. In particular K_S+B_S is \Q-Cartier and its ordinary crepant pullback is the boundary just constructed. This argument does not require K_S itself to be \Q-Cartier.

First take a small \Q-factorialization S_1\to S of the ordinary klt pair (S,B_S). Its crepant boundary is effective and pair discrepancies are unchanged. Now K_{S_1} is \Q-Cartier and S_1 is klt, so underlying-variety discrepancies are defined. Fix

and extract to a \Q-factorial model S'\to S_1 precisely the prime divisors whose underlying log discrepancy is less than \epsilon'. This is a finite set: on a fixed log resolution, write the discrepancy as the nonnegative integral discrepancy of the full reduced SNC support plus a positive linear combination of its integral component orders. A discrepancy below one forces the integral term to vanish. Such valuations are the monomial lc places of the reduced SNC pair, and the positive coefficients bound all their integral weights. There are finitely many strata and finitely many possible weight vectors. The klt extraction furnished by [11] applies since each desired divisor has pair discrepancy at most its underlying discrepancy, hence less than one. An explicit extraction construction is given in Lemma 10.3 below.

For this extraction \pi:S'\to S_1 one has

Thus all remaining underlying discrepancies improve; the extracted primes themselves have underlying discrepancy one on S'. It follows that S' is \epsilon'-lc. The crepant transform B_{S'} of the ordinary pair boundary is effective. If N' denotes the pullback of N, then

Every positive coefficient of N' is at least \delta/2. For an extracted primitive valuation u with u(N)>0, (9.10) gives

Together with (9.8) this yields u(N)\ge\delta-2\epsilon'>\delta/2. For a prime already on S_1, use its corresponding prime on S. Choose a prime of Y dominating it, and let k\ge1 be its fibre multiplicity. The coefficient k u(N) is a positive integer, because E_Y is the birational pushforward of the integral divisor E. Effectivity of (Y,\Delta_Y+H_Y) gives

This proves the assertion for all primes of S'.

Birkar’s effective birationality theorem [10, Theorem 4.2] now applies. Its underlying variety is \epsilon'-lc; its polarization N' is nef and big; N'-K_{S'} is pseudo-effective; and in its required decomposition into an integral pseudo-effective summand and an effective summand we take 0+N'. The positive coefficients of the latter are at least \delta/2. We obtain a positive integer m, taking a common multiple for the finitely many possible positive dimensions of S, such that |\lfloor mN'\rfloor| is birational and in particular nonconstant. Crucially, m depends on neither t nor the denominators of the coefficients of N'.

9.3. The strict ceiling inequality and final mixture

We now fix a rational t>0 with mt<1/20. Pass far enough along the sequence that, writing \epsilon for the prime estimate in Proposition 5.2,

We may enlarge \epsilon slightly to make it rational while retaining this inequality. The section 1 belongs to |\lfloor mN'\rfloor|, and nonconstancy gives another section represented by a nonconstant rational function g on S'. Push its effective divisor \operatorname{div}(g)+mN' to S, pull back by h, and use (9.5). This gives

Let L_0 be the effective big semiample \Q-Cartier divisor on F_0 whose pullback is L. At a prime of F_0 with l_P=0 we have A(P)=1, so its coefficient in the pushforward of E is zero. If l_P>0, its coefficient is

Pushing (9.13) to F_0 therefore gives the effective \Q-Cartier bound

Its pullback controls every prime of F, including the exceptional ones. Set b=1/10. If l_P>0 and n=\ord_P(g)\in\Z, then

Since K_{F,P}\ge-1, this proves \operatorname{div}(g)+K_F+\lceil bL\rceil\ge0 at such a prime. If l_P=0, the pullback bound gives n\ge0, while K_{F,P}\ge0: otherwise K_{F,P}=-1 would be an lc place with l_P=0. The same inequalities with g=1 hold as well. Thus 1 and the nonconstant g are two independent sections of K_F+\lceil bL\rceil, contrary to the assumed dimension one. The positive-dimensional case is impossible, and for this fixed t we have N=0 sufficiently far along the sequence.

Now (Y,\Delta_Y+H_Y) is a klt log Calabi–Yau pair with big boundary, so Y itself is of Fano type. Let J\ge0 with J\sim_{\Q}L and define a form \chi_0 by

Its boundary on Y is the effective divisor \Delta_Y+tJ_Y. Write C_0 for the fixed clipping constant on the set A=0. There, (9.4) and the hypothesis on J give

where (x)_+=\max\{x,0\}. Choose a rational \theta\in(0,1), depending only on C_0, such that

The form with discrepancy A_\chi=(1-\theta)A_\psi+\theta A_{\chi_0} has effective boundary on Y and satisfies A_\chi\ge A_\psi/2 on A_\sigma=0. Theorem 9.1(2), with r=1, C=2 and c=1/2, gives a uniform rational a\in(0,1) such that, putting a_0=a\theta>0,

Since H is effective, we conclude

The final constant is uniform because t was fixed once and for all. This is the asserted integral jump estimate.□

Proof. Choose a smooth projective model T of K and spread out G, taking a dominant projective closure and its normalization. This gives a projective morphism f:X\to T with generic model G. Its Stein factor is finite birational over T because K is algebraically closed in L; normality of T makes it an isomorphism. Hence f has connected fibres.

For the boundary B_\omega defined by \omega, we have

The boundary is effective and sub-lc over the generic point of T. We check the discrepancy rank condition, including the lc case. On a resolution \mu:G'\to G of the generic pair, let K_{G'}+B_{G'}=\mu^*(K_G+B_G). The modified discrepancy divisor omits the components having non-log discrepancy -1, or equivalently boundary coefficient one. At every remaining component its coefficient is the negative of the coefficient of B_{G'}. All coefficients of B_{G'} are at most one, and nonexceptional coefficients lie in [0,1]. Its modified rounded discrepancy divisor is therefore effective and exceptional. If this divisor is denoted by D, normality gives \mu_*\cO_{G'}(D)=\cO_G: sections with poles only at exceptional divisors have no poles at codimension-one points of G. Consequently

This verifies rank one. We have thus checked generic effectivity, generic sub-log-canonicity, rank one, and the rational adjoint pullback in the lc-trivial canonical bundle formula. The b-semiampleness and the identification with the threshold moduli part follow from [5, Theorems 1.5 and 6.28]. On a sufficiently high smooth base, write this formula as

The moduli b-divisor descends to T and its trace is semiample.

To identify the generalized discrepancy and prove attainment, put a primitive base valuation \ord_P on a smooth base and work over the DVR at its generic point. Resolve the boundary and the fibre divisor. On this resolution write

where m_j>0 and the combined support is SNC. If b_j is the coefficient of E_j in B_Z, the threshold is

The supremum is over all real t, since vertical discrepancies may initially be negative. Horizontal coefficients are at most one by the generic hypothesis. At the displayed minimum all coefficients of the resolved SNC pair are at most one; increasing t violates this at a minimizing component. Moreover, m_j^{-1}\ord_{E_j} restricts to \ord_P. Sub-log-canonicity at t_P gives the converse lower bound for every valuation above \ord_P. Hence

This proves both attainment and the claimed generalized discrepancy identity; homogeneity treats nonprimitive base valuations. In particular absolute positivity of A_\omega implies strict positivity of \bar A_\omega, because the latter is an attained minimum.

Perform the same construction for the anchor \psi. Take a common high smooth projective base on which both moduli b-divisors descend and on which the discriminant of \psi has SNC support. Since its moduli part pulls back to every higher model, the ordinary discrepancy of this discriminant subpair is exactly \bar A_\psi on all valuations. Choose a sufficiently divisible q with 1/q\le\gamma and a general member

If this system is trivial, take H=0. Otherwise H is reduced and transverse to all strata of the anchor’s SNC boundary. Adding it with coefficient one leaves that anchor subpair sub-lc, even if some of its boundary coefficients are negative. Thus

The ordinary boundary B'_T=B_{\omega,T}+H/q has discrepancy

which proves (10.3). Equation (10.4) gives K_T+B'_T\sim_{\Q}0. After clearing denominators, write m(K_T+B'_T)=\operatorname{div}(g) and choose a volume form \eta_T with divisor K_T. The form \eta_T^{\otimes m}/g has boundary B'_T, as required. For \omega=\psi and \gamma=1/2, this construction gives

□

Proof. Choose an effective klt log Fano boundary on the initial model. First take a small \Q-factorialization of this ordinary pair. Pair discrepancies and codimension-one boundary coefficients are preserved, and the negative adjoint becomes nef and big. Only after this step do we use the separately \Q-Cartier underlying canonical class. Its underlying variety is klt because its effective pair boundary is klt.

Put \epsilon_0=1/10 and consider the geometric prime valuations whose underlying log discrepancy is less than \epsilon_0. This is a finite set. On a fixed geometric log resolution write K_V+B_V=\rho^*K_X and let D be the full reduced SNC support. Every component has a_i=1-\operatorname{coeff}_{D_i}B_V>0. For a primitive prime valuation v,

The first summand is a nonnegative integer. If A_X(v)<1, it is zero, so v is a monomial lc place of the reduced SNC pair at a stratum. Each nonzero v(D_i) is a positive integer, bounded by 1/a_i. The finitely many strata and weight vectors give finiteness. The same argument works with any fixed threshold strictly below one, as used in Section 9.

The finite set is invariant under the Galois action and can be extracted over Q. To make both the extraction and its effectivity properties explicit, take a high resolution containing the desired divisors and use the chosen effective klt log Fano boundary. At each divisor to be kept, its crepant coefficient is positive and less than one: its pair discrepancy is at most the underlying discrepancy <\epsilon_0. Keep these coefficients and the strict transform of the boundary. At every other exceptional divisor choose a rational coefficient below one but above both zero and its crepant coefficient. The resulting pair is klt, and its adjoint differs from the pullback of the old adjoint by an effective exceptional divisor D_0 whose support does not contain any divisor to be kept.

Run a klt MMP over the old variety to a nef model, using [11]; the morphism is birational, so the boundary is big over that variety. Its relative adjoint is D_0. Negative steps are isomorphisms at the generic points of the divisors to be kept, which lie outside its support. On the final model its transform is effective, exceptional and relatively nef; the negativity lemma makes this transform zero. The resulting \Q-factorial extraction has precisely the required exceptional divisors and carries the crepant effective klt boundary. This extraction construction uses only the effective klt pair and the discrepancy bound at the retained divisors; the log Fano assumption is used for the following positivity. Its negative adjoint is the pullback of the old nef and big negative adjoint, hence nef and big. It is still of Fano type: if -(K+B) is nef and big, write it as A+D up to rational linear equivalence, with A ample and D\ge0; a sufficiently small addition eD preserves klt and has negative adjoint rationally linearly equivalent to (1-e)(-(K+B))+eA, which is ample.

For the underlying varieties, the same extraction satisfies

All nonextracted underlying discrepancies therefore improve, whereas each extracted prime has underlying discrepancy one on X'. This proves geometric \epsilon_0-log-canonicity. The comparison may be made with the entire effective \Q-Cartier exceptional combination B_{\rm exc}, so it does not require its individual geometric components to be defined over Q. A carried effective form boundary likewise remains effective: at an extracted divisor its discrepancy is at most the old underlying discrepancy, and its new coefficient is one minus that discrepancy.

Here are the descent and field-of-definition details. A geometric divisor and the resolution containing it are defined over a finite extension of Q. Passing to a finite separable constant extension at a valuation trivial on Q does not introduce ramification or change the primitive discrepancy. One can thus choose the finite set and its resolution as Galois orbits and carry out the preceding construction over Q. Alternatively, spread the resolution, the boundary and the effective exceptional difference over a sufficiently small base with function field Q, and apply the relative form of [11]; its generic restriction is the same extraction. A small \Q-factorialization on the spread preserves the crepant identities and the geometric generic discrepancies.

Run a K_{X'}-MMP with scaling over Q. Since X' is of Fano type, K_{X'} is not pseudo-effective: its intersection with a sufficiently high power of an ample divisor is negative after writing -K_{X'} as an ample class plus an effective boundary. The klt Mori fibre space theorem of [11] gives a Mori contraction H\to U. The underlying discrepancies improve along this program, also after extending constants, so H remains geometrically \epsilon_0-lc. Every carried effective boundary stays effective by pushforward through the non-extractive map.

The variety H is still of Fano type. Indeed the log Fano structure on X' gives an effective klt log Calabi–Yau boundary that is big, by adding a sufficiently general small-coefficient representative of the ample negative adjoint. Push this boundary through the MMP. Its adjoint is crepant: on a common resolution the difference is exceptional and rationally linearly trivial over the common base, so the negativity lemma makes it zero. Its boundary remains big by pushforward of sections, and the big-boundary perturbation used in Section 9 makes H of Fano type. Over the function field Q, the Mori program can also be obtained by spreading the klt log Fano data over a small open and applying the relative theorem. The canonical class is not pseudo-effective relative to that open, as is seen on its generic fibre. Restriction of the resulting program to the generic fibre preserves all the discrepancy comparisons and yields the stated Mori contraction.

The generic fibre of H\to U is positive-dimensional and has ample negative canonical class. Its discrepancies are computed by primes horizontal over U; after extending the constants of R=Q(U), the preceding geometric comparisons make it \epsilon_0-lc. Thus it is geometrically \epsilon_0-lc Fano. Connectedness of the Mori contraction and characteristic zero make P/R regular. As a subfield of the regular extension P/Q, the extension R/Q is regular as well. The relative dimension of the Mori contraction is positive, so the transcendence degree strictly decreases.

For completeness, U is of Fano type by the following canonical bundle formula argument. Over an algebraic closure of Q, let B_H be an effective klt boundary with A_H=-(K_H+B_H) ample. Fix an ample divisor A_U on U. For sufficiently small rational e>0, A_H-e h^*A_U is ample. Choose a general effective rational representative D_H of this class with small coefficients so that (H,B_H+D_H) is klt. Then

This is a projective connected klt-trivial fibration with effective generic boundary; the rank-one verification in Lemma 10.2 applies. Apply the canonical bundle formula and the moduli-replacement construction from that lemma’s proof to this adjoint pullback. Its threshold discrepancy \bar A is positive at every nonzero divisorial base valuation: the DVR minimum in (10.5) is attained, and all its numerators are positive because the source pair is klt.

On a high smooth \pi:T\to U, let the moduli part M_T descend and be semiample, and let the discriminant B_T have SNC support. Choose compatible representatives as in (10.8), so that

For sufficiently divisible q\ge2, choose a general member H_T\in|qM_T|, taking H_T=0 if the system is trivial. The reduced transverse addition H_T leaves the discriminant subpair sub-lc, including when B_T has negative coefficients. Thus 0\le u(H_T)\le\bar A(u) for every divisorial u, and the discrepancy of B_T+H_T/q lies between \bar A/2 and \bar A. Its pushforward B_U is effective: the discriminant trace is effective by (10.7), and the added divisor is effective. Formula (10.8), with N=-eA_U, gives the \Q-Cartier adjoint and its crepant pullback. Consequently (U,B_U) is klt and K_U+B_U\sim_{\Q}-eA_U, proving that U is of Fano type. The data descend to a finite Galois extension of Q; averaging its conjugate boundaries preserves effectivity, klt and ampleness of the negative adjoint and descends the Fano type structure to Q. The algebraic characteristic-zero canonical bundle formula may be applied here after identifying the algebraic closure of the finitely generated extension of \C with \C as an abstract field and then descending these finite data.

Finally, for every prime of U, a prime of H over its generic point has effective carried boundary. Formula (10.7) gives the final assertion about the effectivity of all substituted forms on this model.□

Proof of Theorem 9.1, assuming Theorem 10.1. For Lifting, induct on n=\operatorname{trdeg}_Q P. If n=0, regularity implies P=Q; take v=u and, for example, \delta=\lambda. If there are no nonzero valuations of Q, the assertion is vacuous.

Suppose n>0 and perform Lemma 10.3. Write R=Q(U) and use \epsilon_0=1/10. Theorem 10.1(1) and Lemma 10.2 give forms \sigma_R,\psi_R such that

Their traces on U are effective, \sigma_R has a uniformly bounded index, and A_{\psi_R}>0. Taking infima in A_\sigma\le C A_\psi and using (10.10), we have

Let \delta_B>0 be the threshold in Theorem 10.1(2) for the generic fibre P/R and target \lambda. If R=Q, that assertion directly proves the desired result. Otherwise apply the induction hypothesis to R/Q, its bounded-index reference \sigma_R, comparison constant 2C, and target \delta_B/2. It supplies a uniform threshold \delta_I. For a given nonzero integral-valued u of Q we have

Indeed z=w|_R is nonzero divisorial, and A_{\psi_R}(z)\le\bar A_\psi(z)\le A_\psi(w). If the right side is below \delta_I, induction gives an integral-valued z of R restricting to a positive multiple of u, with A_{\psi_R}(z)<\delta_B/2. Then (10.10) gives \bar A_\psi(z)<\delta_B, and the bounded-fibre lifting assertion produces the desired integral-valued v of P. Its restriction to Q is again a positive multiple of u. Each tower step lowers n, updates the reference index by a function of already bounded data, and doubles the comparison constant. There are at most d steps. The recursively chosen thresholds therefore depend only on d,r,C,\lambda, proving Lifting with its stated uniformity.

For Mixing, induct on \operatorname{trdeg}_{\C}P. Dimension zero has no nonzero divisorial valuations and is vacuous. Apply Lemma 10.3 over \C, and write R=\C(U). Theorem 10.1(3) gives

on valuations trivial on R and on those restricting to a nonzero u with \bar A_\sigma(u)=0. Together with A_\sigma\le C A_\psi, this gives A_\chi\ge(c-DC)A_\psi there. Choose a rational b\in(0,1), depending only on the stated constants, with b(1+DC)\le1/2, and define

It has effective boundary on H, and on the indicated valuations

If R=\C, this treats every valuation and proves Mixing with a=b.

Otherwise (10.13) makes \chi_+ klt on the generic model over R, since it applies to all valuations trivial on R. Its boundary there is effective, so base substitution applies even if some vertical discrepancies are negative. Obtain \sigma_R,\psi_R as in (10.10), and apply Lemma 10.2 to \chi_+ with the anchor \psi and \gamma=1/4. It gives \chi_R with

All three traces on U are effective. The zero set of A_{\sigma_R} is exactly that of \bar A_\sigma, by (10.10). At a valuation in this set, taking infima in (10.13) gives \bar A_{\chi_+}\ge\bar A_\psi/2 and hence

The lower-dimensional Mixing assertion on the Fano type model U applies with comparison constant 2C and zero-set constant 1/4. Thus a uniform rational a_R\in(0,1) satisfies

For a valuation w of P with nonzero restriction u, we have

by the barred definitions and the upper bounds in (10.10) and (10.14). The same convex mixture is therefore strictly positive upstairs. For valuations trivial on R, positivity follows from (10.13). Finally

The product a=a_Rb is positive, rational and uniform. The dimension decreases at every step, and only the controlled reference index and the displayed constants enter the induction. This proves Mixing and completes the reduction to Theorem 10.1.□

Proof. We use weak toroidalization [2, Theorem 1.1] for projective dominant morphisms in characteristic zero, with prescribed closed subsets, and prove the additional assertion about the logarithmic form. Only projective birational modifications of the source over the resulting base change are needed.

Induction and the relative form. In relative dimension zero, mark the zeros and poles of the relative form function. A dominant generically finite toroidal morphism has invertible logarithmic differential determinant in characteristic zero. This follows in toric charts from the full rank of the map of torus lattices, including the unit directions. More generally that map has full target rank for a dominant toroidal morphism: otherwise the map on logarithmic differentials in the full toric frames would be generically rank-deficient even after completion, contrary to dominance and separability. Faithful flatness of completion transfers this rank test to the original chart.

For the inductive step, choose independent relative rational functions and resolve their graph to factor the morphism projectively through an intermediate base of relative dimension one less. The top arrow then has relative dimension one. In the determinant factorization write \eta=\eta_1\otimes\eta_{\mathrm{low}}, with the appropriate common tensor power understood. Apply weak toroidalization to the top arrow, marking the prescribed sets and the closures of the support of the divisor of \eta_1 on its original normal proper generic curve. A birational normal proper model of that generic curve is the same curve. Consequently every component of the relative logarithmic divisor of \eta_1 which is not already in the boundary is vertical.

Refine this top toroidal arrow so that its base B_1 is smooth strict toroidal, its source is strict simplicial toroidal, and all cone maps are onto cones. Mark on B_1 the images of those finitely many bad vertical components and the entire boundary of B_1. Apply the induction hypothesis to the lower arrow B_1\to B_{\mathrm{old}}. It gives an arrow B_2\to B_{\mathrm{new}}, mapping to the old lower arrow, with the logarithmic form property for \eta_{\mathrm{low}}.

Normalized main pullback. We verify the normalized main pullback of the top arrow under B_2\to B_1. This map need not be toroidal, or dominant on every subsequent parameter. The marked inverse images nevertheless ensure that the old boundary equations pull back to monomials times units in complete toroidal charts. One can see this using the log structure of functions invertible off the boundary. Alternatively, on a strict simplicial chart some multiple of the pull divisor is the divisor of a character monomial. Normality makes the corresponding power of the function that monomial times a unit. At a geometric closed point the unit has a formal root in characteristic zero. The first term for generic positive character weights shows that the monomial exponent itself is divisible by that power, proving the assertion for the original function.

At such a point of the old top arrow, write the boundary coordinates of the smooth base as monomials z^{p_j} times units in the source monoid. The p_j are linearly independent. The smooth base parameters transverse to its boundary can be made part of the smooth source parameter system: the stratum map is smooth, as follows from the full-rank torus map in characteristic zero. First absorb the old source units by a unit-valued character on the source lattice, taking formal roots when its finite lattice index requires them. Scaling monomials by such units is an automorphism of the complete local ring; its action on the monoid generators in the cotangent space is invertible, and completeness gives an inverse by successive approximation. Combined with the preceding smooth-parameter change, the cotangent matrix is invertible. The logarithmic Jacobian of this old source-coordinate change is a unit. Its relative differential need not be zero, but it changes a logarithmic determinant frame by a unit and hence does not change its divisor.

Now pull back to B_2. Absorb the remaining unit factors, which come from the new base, along the old source characters. These multipliers have zero relative differential over B_2. Substitute for the old smooth base parameters, retaining all the smooth parameters of B_2. The remaining equations are binomial:

Their normalized main branches are toroidal, with cone the intersection of the product cones with

This intersection meets the relative interior of the product cone: the first cone maps onto the old base cone, and the relative interior of the second maps into the relative interior of its relevant old base cone. Inverting all monomials in (11.4) gives the character-group pushout. Its finite torsion, when present, separates the torus branches. The closure of each such branch is an affine monoid variety; its normalization is the saturation of that monoid in its character lattice. The interior condition makes its monoid sharp and gives a unique point over the monoid origin on each branch. Completion at that point is one normal monoid-series domain, with the free smooth parameters adjoined.

These are the branches of the actual algebraic main pullback. The joint nonboundary open is dense in that main component, so boundary monomials are nonzerodivisors there. Flat completion preserves this property; no minimal completed branch can be supported solely on vanishing boundary monomials. Thus the relevant completed branches are precisely the closures of the torus branches just computed. Excellence gives finite normalization and its compatibility with this completion; geometric generic integrality identifies the prescribed main generic component. These facts also justify performing the binomial computation first in polynomial monoid algebras and then completing. They do not require an arbitrary formal coordinate change to be algebraic.

The chart map to B_2 retains its given character map and smooth parameters. Its cone is onto the B_2 cone, since the old top cone was onto. Characters rationally complementary to the p_j, together with the remaining relative smooth parameters, give the same relative logarithmic determinant frame after pullback: saturation changes no rational span, and the new unit multipliers contribute only base differentials. It follows that the relative logarithmic determinant pulls back unchanged. The divisor of \eta_1 is now wholly supported in the boundary, because its bad vertical images were marked. Composing with B_2\to B_{\mathrm{new}} gives a toroidal map; in these same charts its base equations are monomials, its log rank is full, and its stratum coordinates are smooth. The relative determinant factorization proves the assertion for \eta. The models continue to map birationally to the old ones and retain all inverse marks.

Projective cone refinement. We carry out these refinements on the finite-type packages over \C, before evaluating at field-valued parameter points. First take compatible projective toroidal resolutions, so that the source and base are smooth and strict. On the resulting regular source cone complex, assign value 1 to each primitive ray generator and extend linearly on each cone. These functions agree on common faces and define a slicing function h, positive away from the origins. Projective equidimensionalization [3, Proposition 4.4 and Lemma 4.1] then gives a smooth base and makes every source cone map onto a base cone. Retain h on the source subdivision: every new cone lies in an old cone, so h remains linear there and positive away from the origins. Apply [1, Corollary 0.11] to its 1-skeleton with the unchanged triangulation induced by the zero lifting function. This gives a projective triangulation with no new rays. Since the base is simplicial, each old source ray maps to a base ray or to zero [3, Remark 4.2]; the image of each new simplicial cone is therefore a face of a base cone. The final source is strict and simplicial, and the cone-surjectivity condition is preserved. All these are toroidal modifications, so the inverse marks and the pullback identities for the logarithmic divisors are retained.

Return to all parameters. Apply the construction to the compactified families, and use the all-parameter shrinking and induction above. Any loci newly excluded by these refinements undergo the same noetherian restart before the finite package list and its common moduli multiple are fixed. On the good isomorphism open the parameter map already lifts to the modified base; properness then extends a valuation-ring map when required. The retained inverse maps and form identities give the claimed birational and differential assertions on every fibre used.□

Proof. First compute the excess using finitely generated fields. After a finite base extension by a root uniformizer \widetilde\pi, clear the ratios w(z_i)/u(\pi) and put z'_i=\widetilde\pi^{-h_i}z_i, of value zero. Prolong all valuations with the old scale. Form discrepancies are unchanged under this finite extension. On the smaller rational field in the variables z' over the extended base field, extract the prolonged base prime and take, near its generic point, a smooth base model times the smooth unit torus with coordinates z'_i. Denote this model by T, its reduced base prime pullback by P_0, and the prolonged restriction of w by v. Then

To check the equality, use the finite-extension discrepancy formula for L/K(z), the ratio (12.5), and invariance of the relative d\log z determinant under rescaling by base functions. They reduce the absolute form to \zeta\wedge d\log z'_1\wedge\cdots\wedge d\log z'_n. At the base prime, the divisor of \zeta contributes exactly A_\zeta(u) after the discrepancy of the reduced-fibre pair is separated off. The contribution of F is \phi(x). This calculation uses the homogeneous scale throughout and proves (TE). The smooth reduced pair (T,P_0) is lc, so E(w)\ge0; for w_x, the established Gauss rule is the order of P_0 with its scale, giving E(w_x)=0. Every cone option occurs, and the minimum defining \mu(b) has a rational minimizer with a label. Minimizing (TE) therefore proves (T), including for the universal generic family.

We now prove (TV). The combined degree of the generic zero and pole divisors of H, with multiplicities, is at most C_1p. This remains a uniform bound after the preliminary bounded Kummer preparation. Its normalized models are finite over the old base pullbacks; over the good opens they give the prescribed scalar-extended generic models mapping to G. Pulling back the bounded polynomial cuts on G bounds degrees in fixed projective embeddings of these finitely many families. Intersection degrees of fixed line bundles in finite type are bounded after stratifying by flatness. The generic fibres used are geometrically normal, both by preparation and, off the base boundary, by eliminating smooth parameters in the toroidal charts.

For the first inequality take the root uniformizer large enough that x/u(\widetilde\pi)\in N_\delta. This auxiliary root degree is allowed to be unbounded. If \pi=\rho^d is the chosen extension, the new ray has primitive lattice vector

Its last coordinate proves primitivity and the displayed splitting. Thus the open ray chart is a torus over \kappa[\rho], with \rho a regular coordinate vanishing to order one. Including the free smooth parameters gives a regular chart with a single reduced smooth vertical divisor.

Make the algebraic cut-ideal modification over the new discretely valued function field, and follow the prolonged w. Every equal-cut ratio has value zero; in a blowup chart the ratio or its inverse is regular, so it is a unit at the centre. Imposing all these unit conditions, and removing boundary components outside the supporting face, puts the centre in this open ray locus. This remains true when smooth parameters vanish additionally at the centre: the open ray chart, with its smooth parameters, is still regular. The vertical prime is the prime with the label already specified, and its scaled order agrees with w_x by (12.4). The rescaled boundary functions z'_i are units: a suitable power is a toric character of zero ray order times a local unit, and normality removes that power. The chosen free stratum units remain units as well.

These assertions hold on the algebraic trait, not only on its geometric completion. The map from its excellent characteristic-zero valuation DVR to the completed trait with algebraically closed residue is flat and regular, with the same uniformizer. Regular base change preserves normality. Geometric generic integrality and flatness identify the main component, so finite main normalization commutes with this base change. Blowups commute with flat base change, and the same normality argument applies to their finite normalizations. Smoothness and a reduced smooth special fibre may consequently be checked in the geometric completed charts and descend at the centre. The resulting local vertical divisor downstairs has the same uniformizer-normalized order.

In this regular local ring factor the principal divisor of H into its vertical order and its horizontal prime factors. The vertical part gives w_x(H). Thus w(H)-w_x(H) is a signed sum of the orders of effective local Cartier horizontal primes D, with their zero or pole multiplicities. Their total weighted geometric generic degree is still at most C_1p. The modifications have changed no generic fibre. Under the possibly unbounded scalar extension the degrees of split components sum, with multiplicities, to the original geometric degree. Rescaling the z_i by base constants changes coefficients, not their rational presentation degrees. Thus this degree budget has no dependence on the auxiliary root degree.

For each horizontal D, choose a nonzero polynomial q(z) vanishing on its generic image with

The z-projection is defined at the generic point of D, since the rescaled coordinates are regular units at our centre and their chosen presentations have nonvanishing denominators on this neighbourhood. Its image closure has dimension at most n-1. Bezout, applied to the bounded-degree presentations of z, bounds its degree by C\deg D, even when the image is contracted. A subvariety of that degree in affine n-space lies in a nonzero hypersurface of at most that degree, by generic linear projection. This proves (12.8) over the actual extended base field. Rewrite q in z' and multiply by a base constant so that its minimum coefficient valuation is zero. It is regular at the centre and its reduction is a nonzero polynomial. It still vanishes along D, whence

On a smooth torus over a characteristic-zero field, a nonzero polynomial of degree d_q>0 has log canonical threshold at least 1/d_q. One may extend the field algebraically and prove this local assertion at each zero as follows. Choose successive general affine linear sections through the point, ending in a line on which the polynomial is not identically zero. Its vanishing order on that line is at most d_q, so the line pair with coefficient 1/d_q is lc. Smooth-divisor inversion of adjunction [26], applied successively and then dropping the added section divisor, gives log canonicity in the ambient smooth space near the point. Points where the polynomial is a unit impose no condition. A nonzero constant likewise contributes zero order.

Apply this bound to the reduction of the normalized q. Inversion of adjunction along the smooth reduced fibre P_0 gives

It suffices to apply this over the generic point of the extracted base prime, where all centres in use lie. Spreading its residue-field resolution, or spreading the polynomial lc assertion over a base neighbourhood, verifies the hypotheses there. Combining (12.8)–(12.10) and summing with the zero and pole multiplicities proves the first inequality of (TV).

For the second inequality keep one logarithmic coordinate chart and its label along the segment. There is a polynomial relation

Indeed, z is generically finite by differential independence. Take the equation of its irreducible graph image in \mathbb P^n_z\times\mathbb P^1_H, then dehomogenize. Its degree in the z-block is counted by n-1 general z-hyperplanes and one general H-hyperplane. These intersections can be taken in the rational-map open over which the generic map to the graph image is finite. The z cuts have bounded degree and the general H cut has degree at most C_1p, bounded by its pole divisor. Bezout in the fixed bounded embedding proves (12.11). The chosen chart-presentation open is dense on the generic fibre and its image contains a dense open of the graph image, so it suffices for this computation. For constant H the assertion is immediate.

By the Gauss rule over the fixed u, the value of each nonzero a_j(z) is a finite minimum of affine functions on the segment, with Lipschitz constant at most C_5p. In (12.11) the least term value must be attained at least twice. Hence w_x(H) lies among the finitely many candidates

each piecewise affine with Lipschitz constant at most 2C_5p. The actual value is continuous on the closed labelled segment by the monoid-series calculation. Partition the segment at the finitely many affine breakpoints and intersections of these candidates. On each remaining interval a continuous selection follows one of the affine candidates, and therefore has the same Lipschitz bound. Adding lengths gives that bound on the whole segment, including its labelled endpoints. Enlarge C_2 to dominate the constants in the two inequalities. This proves (TV).□

Proof. Choose P smooth with strict SNC toroidal boundary, and let C_P^\natural be the crepant transform of C^\natural. Set

Then (P,D) is effective dlt, N\geq0, and

The lc places of the two pairs agree. To check the nontrivial direction, the centres of lc places of the SNC pair (P,D) are intersections of coefficient-one components, and none is contained in the components truncated in (13.4). Every such place has v(N)=0. Conversely A_{C^\natural}=0 forces both nonnegative terms on the right of (13.5) to vanish. On the generic fibre N is exceptional over G_{\rm gen}, since the form boundary on that model is effective.

We construct an MMP with scaling, give the scaling alternative, and then show that the exceptional error vanishes on a finite model. All the varieties used here are of finite type over \kappa. If needed for the ordinary complex statements of the MMP theorems, transport the entire family by an abstract isomorphism \kappa\simeq\C. Such an isomorphism exists because \kappa is the algebraic closure of a finitely generated extension of \C and has the same transcendence degree over \Q as \C. This identification need not fix the original copy of \C.

Choose a rational ample divisor H with K_P+D+H nef over B. Write P_i,D_i,N_i,H_i for successive models and pushforwards, and put C_i^\natural=D_i-N_i. These subpairs are crepant throughout: the adjoint of C^\natural is rationally linearly pulled back from the base, so, with compatible canonical divisors, its pullbacks to a common resolution agree. For each negative MMP step, with common resolution maps a and a^+, negativity therefore gives

All N_i remain effective. In particular K_{P_i}+D_i\sim_{\Q,B}N_i is relatively pseudo-effective, so a Mori fibre ending is impossible.

Scaling construction. Here are the existence and scaling details. For any rational 0<t\leq1, choose a sufficiently small rational \epsilon>0. The divisor tH+\epsilon D is ample, and a sufficiently divisible general representative T_{t,\epsilon} gives an effective big klt boundary

The general representative has arbitrarily small coefficients and meets the SNC support transversely; (P,(1-\epsilon)D) is klt. If the steps so far have scaling thresholds at least t, they are nonpositive for K_P+\Delta_t. Hence its pushed boundary is still klt, big, and rationally linearly equivalent to D_i+tH_i. Bigness is preserved under these non-extractive birational maps. On any such model a klt big boundary can moreover be represented with a small general ample part. Indeed take an effective big decomposition with the required ample part and mix it, with sufficiently small positive coefficient, into the existing klt representative.

Suppose that K_{P_i}+D_i is not nef, while K_{P_i}+D_i+\lambda_{i-1}H_i is nef. Its nef threshold \lambda_i on this interval is positive. Choose a rational t\in(0,\lambda_i). The klt cone theorem applied to a representative at t with ample part leaves only finitely many negative extremal rays to test. The remaining part of the cone is nonnegative both at t and at \lambda_{i-1}, hence throughout their interval. Thus \lambda_i is a rational maximum of finitely many ray thresholds, and is attained on a ray R_i with

The klt contraction theorem at t contracts R_i. At \lambda_i the nef divisor is relatively semiample: use its klt representative with an ample part and apply basepoint-freeness to that representative with the ample part removed. It therefore descends through the contraction. If the contraction is small, the klt flip supplied by [11] at t is also the flip for K_{P_i}+D_i: over the contraction their classes are positive multiples, because the \lambda_i combination descends. These are the ordinary dlt MMP steps; they preserve \Q-factoriality and dlt, and are nonpositive for all the positive-scaling combinations with parameter at most \lambda_i. This constructs the sequence until nefness, or constructs an infinite sequence with nonincreasing thresholds.

Limiting threshold. We next prove that an infinite sequence must have

Every divisorial contraction lowers the relative Picard number, so there are only finitely many. Fix an index j after the last one. If the thresholds had positive limit, choose a rational closed interval bounded away from zero, with upper endpoint at most the previous scaling bound, containing all sufficiently late thresholds. On P_j, choose klt big representatives of D_j+tH_j at the two endpoints. They can be chosen to contain small common positive multiples of finitely many general ample divisors spanning N^1(P_j/B), as well as a fixed ample part. To obtain these supports, subtract the desired sufficiently small ample sum from the big class, represent the remainder effectively, and mix this representative into a klt one with a small coefficient. The endpoint representatives determine a rational segment of klt boundaries. Allowing small independent changes in the coefficients of the spanning divisors embeds it in a rational polytope of klt boundaries with fixed ample part. The adjoints remain big after shrinking these changes: on the segment their relative classes are N_j+tH_j, an effective class plus a big class with t>0.

BCHM finiteness for this polytope gives finitely many marked ample models over B [11, Corollary 1.1.5]. Every sufficiently late P_i is one of them. In fact P_j\dashrightarrow P_i is small, and the pushes of the chosen divisor classes still span N^1(P_i/B). For completeness, a small birational map between \Q-factorial models identifies these numerical spaces: the pullback difference of a numerically trivial divisor and its transform on a common resolution is exceptional over the other model and numerically trivial on its contracted curves; negativity applied with both signs makes that difference zero. The same argument in reverse proves the assertion. At the parameter \lambda_i the pushed adjoint is nef. A sufficiently small rational change in the spanning coefficient directions adds a relatively ample class on P_i, while staying in the polytope, so the resulting adjoint is ample there. Sections of the corresponding divisors agree across the small map, as they are determined by the same inequalities at prime divisors. Thus P_i, with its marking from P_j, is the ample model of this perturbed boundary.

Distinct negative steps cannot return to the same marked model. The pullback difference for K+D is effective at each step and is nonzero at a genuinely negative step. If it were zero, the two pullbacks would agree on a common resolution. A curve dominating a negative contracted curve would then have negative intersection on the old side, whereas its image on the other side is contracted to the same contraction base and has nonnegative intersection there, a contradiction. This gives strict discrepancy improvement at some valuation. Subsequent discrepancies cannot decrease, so repetition of the marked pair is impossible. The finite list of marked ample models therefore rules out a positive limiting threshold, proving (13.8).

Preservation of lc centres. The lc centres and their incidences have been retained at every step. Indeed discrepancy monotonicity and crepancy give

Together with the equality of initial lc places, this shows that the lc places are exactly the same and have v(N_i)=0. Their centres are consequently not contained in \Supp N_i. Every negative contracted curve is contained in \Supp N_i, since an effective \Q-Cartier divisor has nonnegative intersection with a curve not contained in its support. The contraction is therefore an isomorphism over a target neighbourhood of the image of each lc centre’s generic point. To see this, a positive-dimensional projective fibre through a point outside \Supp N_i would contain a contracted curve through that point. A zero-dimensional point of a connected fibre cannot be a separate component of a larger fibre. Properness and normality of the target then give the asserted isomorphism neighbourhood. The flip agrees over the same neighbourhood. For an inclusion of lc centres, test it at the generic point of the smaller centre; this lies in that common isomorphism locus. Inclusions are thus preserved in both directions, and properness preserves their images in B. Only finite compositions will be used below.

Movable representatives. Fix j after the last divisorial contraction in a putative infinite sequence, or take its final nef model in the finite case. Fix also a relatively ample Cartier divisor A on P_j. We claim that for every rational \delta>0 and every finite list of primes on P_j there is an effective divisor avoiding that list with

In the finite case N_j is nef and the sum is ample, so general representatives on the affine base give the claim. In the infinite case choose i first, sufficiently large that \delta A-\lambda_iH_j is ample on the fixed model P_j. All maps from P_j to P_i are small. Choose an ample Cartier divisor A_i^+ on P_i. Once i and this divisor have been fixed, openness of the ample cone permits a positive rational \alpha such that

The divisor N_i+\lambda_iH_i+\alpha A_i^+ is also ample, being nef plus ample. On the affine base choose effective representatives of it and of (13.10), avoiding the corresponding finite lists of primes. Transform the first back to P_j and add the second. Smallness identifies the primes and yields (13.9). The choice of \alpha is made after i; no bound on the size of (A_i^+)_{P_j} is required.

Vanishing of the exceptional error. We show first that N_j is vertical over B. On the generic fibre take a common resolution of P, P_j, and G_{\rm gen}. The effective Cartier pullbacks of N_j are bounded above by those of N by (13.6). Since N is exceptional over G_{\rm gen}, the order of N_j at every prime of G_{\rm gen} is therefore zero. This assertion concerns Cartier pullback data on a common resolution; it does not assert a morphism P_j\to G_{\rm gen}.

Suppose a horizontal component F of N_j has coefficient n_F>0. Apply (13.9) with F in the avoided list. On the generic fibre relative linear equivalence becomes ordinary linear equivalence, so, for a rational function h_\delta and a positive integer m_\delta,

Taking its coefficient at F gives, with a_F=\ord_F(A),

Pull this effective divisor to the common resolution and take its trace on G_{\rm gen}. The zero trace of N_j gives

where A_{\rm tr} is the fixed Weil trace of the Cartier pullback data of A. Choose a Cartier divisor H_G\geq A_{\rm tr} on the normal projective variety G_{\rm gen}. Such a divisor exists: for a sufficiently large multiple of an ample Cartier divisor L_G, the coherent divisorial sheaf \mathcal O_{G_{\rm gen}}(mL_G-A_{\rm tr}) has a nonzero section h, and mL_G+\operatorname{div}(h) is a Cartier majorant. Consequently the divisor obtained from (13.12) by replacing A_{\rm tr} with H_G is effective and \Q-Cartier. Pulling this divisor to the valuation F gives

Together with (13.11), this implies

for arbitrarily small positive rational \delta, a contradiction. This proves verticality without assuming G_{\rm gen} is \Q-factorial or pulling back a Weil divisor that is not Cartier.

Next N_j is very exceptional over B: it is vertical, and over every prime divisor T\subset B some prime divisor dominating T is absent from its support. A component of N_j dominating a base prime must dominate a boundary prime, because it is the transform of a toroidal component of N. For a boundary prime T, minimize \phi over a positive vector on its base axis. A rational minimizer exists, and its supporting face has g=0. Its toroidal divisorial valuation has an lc centre dominating T on P, hence on P_j. That centre is not contained in \Supp N_j. The pullback of a local Cartier equation of T vanishes at its generic point. A prime component of this Cartier pullback containing that centre dominates T and is absent from N_j. This reasoning also applies when the lc centre has higher codimension; the centre itself need not be a prime divisor. For a non-boundary prime T, surjectivity and the same Cartier pullback argument supply a dominating prime, and no component of N_j dominates T.

Choose rational \delta_k\downarrow0 and representatives in (13.9) avoiding all components of N_j. For each such component F, their restrictions to F are effective Cartier divisors after clearing denominators, with proper closed supports. Every curve in F over B not contained in the countable union of these supports satisfies

This is precisely nefness on very general curves of F/B. The morphism P_j\to B is still a projective contraction: its generic function field is regular over that of the normal base, so its finite Stein factor is B. Apply Shokurov’s very-exceptional negativity lemma in the form of [6, Lemma 3.3] to

Its negative part is very exceptional, and -D'=N_j is nef on the required very general curves. The lemma gives D'\geq0. Since N_j\geq0, it follows that N_j=0.

This vanishing occurs on the fixed finite model P_j. It makes K_{P_j}+D_j relatively rationally linearly trivial and rules out any subsequent negative step. In particular the proposed infinite sequence cannot occur. The effective dlt structure and the asserted centre correspondence are therefore obtained after finitely many steps, as claimed.□

Proof. First disregard V and C_s. A minimizer for \phi over a rational vector in \relint(\tau_s) has a support face on which g vanishes. The onto-cone refinements imply that this face maps onto \tau_s. Its stratum occurs over s: the smooth stratum map has dense open image, properness gives a specialization over s, and the toroidal face chart at that specialization has open stratum dense in its ordinary fibre branch. This is also the stratum availability used in (T). Since Y_s is connected, this proves the assertion when V=Y.

Assume \delta_0\ne0. Closed strata in this strict toroidal model are normal and are unions of strata. At a deeper stratum the boundary divisors containing V determine a unique face and a unique normal orbit-closure germ. Strictness identifies these divisors globally, so different local face branches of the same closed stratum cannot be confused. In particular their individual ray directions, and coefficients on those directions, agree at incidences.

We separate the chosen component by a pointed étale neighbourhood

Indeed the Stein factor S_V of the proper map V\to B is finite over B, also when V\to B is nondominant. The points of its fibre over s correspond to the connected components of V_s. Over the henselization at s, the finite algebra splits by the idempotents separating these points. The finitely many idempotents descend to some pointed étale neighbourhood. The associated open-and-closed piece of V\times_B B_2 has precisely C_s in its pointed fibre. This pullback is normal, and its irreducible components are disjoint. Thus C_s lies in one such component V_2, whose fibre at s_2 is exactly C_s as a topological space. The component V_2 is again a closed stratum.

Shrink to a connected smooth affine neighbourhood of s_2. The pulled-back Y remains integral: its generic fibre is geometrically integral and its normal étale pullback has no additional vertical component. It remains strict toroidal with connected fibres. The forms, cones, and gap function pull back, and the pointed ordinary fibres are identified. Use the notation Y/B,s for this new family during the existence argument. No lift of the trait to B_2 is needed, and no bound on the degree of this neighbourhood will be used.

Apply Lemma 13.1 to obtain the effective dlt crepant contraction (P_j,D_j)\to B. On its initial toroidal resolution P, one lc centre maps onto V_2: take the centre of a toroidal lc valuation with weights in \relint(\delta_0). Another lc centre maps onto the closure of an all-zero cone over \tau_s furnished by the first paragraph. The latter centre has base image \overline{O_{\tau_s}}. Transform both to P_j. Inside the second centre choose an lc centre Z minimal by inclusion among those whose image contains s. This is also inclusion-minimal among all lc centres with image containing s: any strictly smaller one would be a subcentre of the second centre as well.

Kollár’s linking theorem [27, Theorem 10] now applies. Its hypotheses hold: the morphism is proper with connected fibre over s, the boundary is effective, the pair is dlt, and K_{P_j}+D_j\sim_{\Q,B}0. It yields inside the first centre a subcentre Z' whose image contains s and which is \mathbb P^1-linked to Z. Directly linked centres have the same base image, by the definition of a link [27, Definition 9]; hence so do centres joined by a chain of links. Consequently

Return these centre inclusions through the finite MMP to P, using Lemma 13.1, and then map to Y. On the SNC model the centres are intersections of coefficient-one divisors, so are closures of all-zero strata. A toroidal subdivision maps such a closure to a stratum closure with g=0: a relative interior zero vector lies in its old support cone, and nonnegativity and linearity force g to vanish on that cone. We obtain an all-zero closed stratum contained in V_2, whose image contains s and is contained in \overline{O_{\tau_s}}. Its image is itself the closure of a base stratum, by smoothness of the open stratum map and properness. Since s\in O_{\tau_s}, these two containments force its image to equal \overline{O_{\tau_s}}.

Its cone therefore maps onto \tau_s, and its open stratum meets the ordinary fibre over s. Explicitly, at a closure point in that fibre the relevant face branch already kills every base boundary equation; the remaining base parameters are smooth stratum parameters and can be eliminated. The open stratum is dense in this fibre branch, so it supplies the required point. The point belongs to (V_2)_s=C_s. Projecting through the pointed étale neighbourhood returns an all-zero stratum and this component incidence to the original ordinary fibre. This proves the proposition.□