Relative denominators and effective systems for log Calabi-Yau fibrations

1 Introduction

Let (X,B) be a projective log canonical pair over \C, and let f:X\to Z be a contraction: a surjective projective morphism of normal projective varieties with f_*\OO_X=\OO_Z. We consider the case

where D is a rational Cartier divisor on Z. The adjoint of the generic fibre is then rationally linearly trivial. The canonical bundle formula separates the contribution of singular fibres from a moduli part on the base. Its qualitative form places that moduli part on a birational model as a nef rational Cartier divisor. For effective applications, one needs a Cartier multiple controlled by the dimension and boundary coefficients, rather than by the individual fibration.

This is a question about a chosen rational presentation. Replacing a base divisor by an arbitrary rationally linearly equivalent divisor changes the moduli part by a rational principal divisor and can introduce arbitrarily large denominators. We prove that, when \dim X\leq4 and the coefficients of B belong to a fixed finite rational set, one can choose an exact pullback presentation whose moduli part has a uniform Cartier denominator.

We recall the base data in the statement. For the given f, the discriminant B_Z has coefficient 1-t_P at a prime divisor P\subset Z, where t_P is the log canonical threshold of f^*P over the generic point of P. This is computed on a neighbourhood of that point where P is Cartier. The moduli data form a rational b-divisor \mathbf M. On a birational model q:W\to Z where it descends, it is represented by a rational Cartier divisor M_W and its pullbacks to higher models; its trace on Z is M_Z=q_*M_W. The b-divisor is b-nef if M_W is nef, and p\mathbf M is b-Cartier if pM_W is Cartier. A generalized pair (Z,B_Z+M_Z) has K_Z+B_Z+M_Z rational Cartier, and its crepant boundaries are defined by

and by the same rule on higher models. It is generalized lc if all coefficients of these boundaries are at most one, and generalized klt if they are strictly less than one. The crepant boundaries on higher models may have negative coefficients.

The first equality in (1.1) is an equality of actual rational divisors, for compatible choices of canonical divisors and the displayed rational function. It selects the representative D_Z\sim_\Q D to which the moduli bound applies. The two integers have distinct roles: p_0 gives the principal correction and, on the generic fibre, a trivialization in that degree; p clears the moduli trace on a model determining \mathbf M. The discriminant is controlled instead by the fixed DCC set \mathcal B.

1.1 Denominators, positivity, and earlier work

Fujino and Mori bound the denominator of the semistable divisor accompanying bK_X in their canonical bundle formula, where b is the least nonvanishing pluricanonical degree of the generic fibre. Their bound uses the middle Betti number of a smooth model of a cyclic cover of the fibre of Kodaira dimension zero. The proof reduces to a curve and controls a finite character through this cohomology [12, Theorem 3.1 and Sections 3.3, 3.6–3.8]. In their klt log Iitaka setting, Todorov and Xu obtain a bound for the moduli denominator in terms of the boundary coefficients in relative dimension two, with arbitrary total dimension [21, Theorem 1.3]. Their proof first controls surface indices and the second Betti number of a resolution of the cyclic cover, then applies the Fujino–Mori character method.

When the generic fibre is a rational curve, Floris bounds a common integer clearing all moduli coefficients in terms of the fibre Cartier index. Her examples rule out every polynomial bound for such an integer as the index varies [9, Theorem 1.6]. For the lc total spaces of dimension at most four considered here, the curve reduction produces reduced special fibres of dimension at most three. The index of the whole reduced fibre supplies the source of uniformity in our proof.

Qualitative moduli theory supplies a different part of the argument. Ambro establishes rational b-Cartier descent and positivity in the klt-trivial setting [1, 2], and Fujino and Gongyo extend b-nefness to lc-trivial fibrations [11, Theorem 3.6]. These results put the moduli data on a suitable base model and make the determining divisor nef; their Cartier multiple may depend on the fibration. We use this descent and nefness after fixing the exact rational trivialization and verifying the required rank-one condition, and obtain a common multiple by a separate special-fibre calculation. The preprint of Bakker, Filipazzi, Mauri and Tsimerman proves qualitative b-semiampleness for lc-trivial fibrations with generically effective boundary [3, Theorem 1.5]. Their Section 7.2.4 discusses the general effective question and known special cases, where one seeks a uniform multiple that is base point free on a determining model. A bound for the Cartier denominator makes a multiple Cartier on such a model; base point freeness is an additional condition, and is not used here.

The argument on a reducible fibre also has an established ancestry. Fujino’s admissible and preadmissible sections organize pluricanonical descent across the conductor [10, Definition 4.1 and Lemma 4.2]. Gongyo uses this framework in the abundance theorem for numerically trivial slc adjoints [13, Theorem 1.5 and Section 5]. Jiang and Liu supply the uniform index theorem for projective slc log Calabi–Yau pairs through dimension three that we use below [16, Corollary 1.6, arXiv version 1]. The relevant output is one trivialization on the slc pair itself, with its conductor gluing. It is this global form that permits the cyclic-character comparison in the present proof.

1.2 The route to the relative bound

Section 3 applies the lower-dimensional index theorem to the geometric generic fibre, whose dimension is at most three, and descends its principalization by Hilbert 90 in the same degree p_0. This fixes the exact presentation (1.1). Threshold ACC supplies the DCC set for B_Z. After applying the qualitative canonical bundle formula and passing to a smooth determining model W\to Z, it remains to control one moduli coefficient at a time. For a prime P\subset W, let \alpha be the coefficient of the pullback of D_Z, and let t_P be the threshold computed from the crepant sub-pair on a resolved diagram over W. The coefficient of M_W differs from \alpha+t_P by an integer. Section 4 takes a transverse curve slice and runs a relative dlt MMP to obtain an exact local identity on a dlt model. Adjunction in Section 5 then gives an slc pair (T,B_T) on the whole connected reduced special fibre, with \dim T\leq3 and K_T+B_T\sim_\Q0.

The decisive uniform input is a form on this entire fibre. Global ACC places the different coefficients in a fixed finite set. The Jiang–Liu index theorem then gives a nowhere-vanishing log pluricanonical form u on T in one even multiple p of p_0. The local residue test compares its pullback with the residues of an ambient form on a normalized cyclic cover; that ambient form carries the character recording the remaining denominator. The comparison may use a larger, pair-dependent degree for ambient adjunction, but it yields one scalar on each normalized component of the connected reduced covering fibre T_Y. In the uniform degree p, the local conductor calculation shows that each tuple has matching nonzero next residues at a generic crossing of two components. Their scalars therefore agree across intersections and hence on all of T_Y. The ambient tuple is therefore one scalar multiple of the invariant pullback of u, even when the cyclic group permutes components. Its character is trivial in degree p, giving p(\alpha+t_P)\in\Z and the required Cartier bound.

1.3 Effective systems and torsion

When D is big, Proposition 6.1 gives an integer m=m(d,\Phi) such that, for every positive multiple l of m, the complete rounded divisorial system

is nonempty and its section ratios generate exactly \C(Z)\subset\C(X). Here the sheaf is the rank-one reflexive divisorial sheaf. Equality of the embedded fields recovers the finite part of the fibration as well as the dimension of its image.

For every positive l divisible by p_0, Section 6 uses the exact presentation to identify the full section spaces inside \C(X):

where f^* means pullback of rational functions. The valuation argument works for reflexive sheaves without requiring l(K_X+B) to be Cartier. Birkar–Zhang’s polarized effective birationality theorem applies to the resulting big lc adjoint on W, with fixed DCC boundary coefficients and pM_W nef Cartier [6, Theorem 1.3]. Choosing m divisible by p_0 and their uniform degrees, the equality above transfers the full base field to every required source system: the common factor cancels in ratios. Their separate Iitaka-fibration bound [6, Theorem 1.2] retains dependence on a fibre nonvanishing degree and a Betti number of a cyclic fibre cover.

Proposition 7.1 is a separate torsion result. Fix 1\leq e\leq4, a rational DCC set I\subset[0,1], and a positive integer p. For projective generalized klt pairs (Z,B_Z+M_Z) of dimension e, with boundary coefficients in I, b-nef rational data \mathbf M for which p\mathbf M is b-Cartier, and a rationally connected smooth projective resolution of Z, it gives one integer \ell=\ell(e,I,p) such that

Using suitable extractions, global ACC places the boundary coefficients in a fixed finite set and gives a uniform positive lower bound for generalized log discrepancies. Birkar’s boundedness theorem [5, Theorem 1.7] bounds the varieties up to isomorphism in codimension one. The proof combines their smooth-locus topology with Kummer theory and finite generation of the divisor class group \Cl(Z) from the rationally connected resolution to give a uniform exponent for torsion in \Cl(Z). A separate uniform multiple clears the coefficients of the actual adjoint before that torsion bound is applied. Birkar’s ordinary-pair index consequence, observed by Totaro [5, Corollary 1.8 and Section 9.5], is a close predecessor.

Corollary 7.2 combines this result with Theorem 1.1. A projective klt fourfold pair with fixed finite rational boundary coefficients and K_X+B\sim_\Q0 has a uniform integral principal multiple of K_X+B whenever it admits a contraction to a positive-dimensional base with a rationally connected smooth projective resolution. The statement includes B=0.

2 Divisors and the lower-dimensional index input

We fix the divisor conventions needed for the exact pullback identities, the rounded systems, and the index theorem on the reduced fibre. All varieties are over \C. Unless stated otherwise, they are integral and normal. In the global statements, all varieties are projective, including the bases of the contractions. The reduced fibres may be reducible and nonnormal, and their normalizations may have several connected components. In the local curve arguments, the total spaces are projective over the indicated curves. After a curve is shrunk to an open neighbourhood, neither it nor its total space need be projective over \C. A contraction is a surjective projective morphism f with f_*\OO=\OO.

Boundaries are effective rational divisors; crepant sub-boundaries may have negative coefficients. We use the usual discrepancy conventions for lc, klt, and dlt pairs. A reduced pair (T,B_T) is slc if T is pure-dimensional, S_2, and nodal in codimension one, no component of B_T is contained in the conductor, K_T+B_T is rational Cartier, and the normalization with the conductor included in the boundary is lc. In particular, the conductor has coefficient one on each normalized branch. The fibres to which we apply this definition will be proved to have the required pure dimension and depth properties.

Canonical divisors are chosen using rational top forms, and we retain those choices in exact divisor identities. For rational divisors on a normal variety, D_1\sim_\Q D_2 means that some positive integer multiple of D_1-D_2 is the divisor of a rational function. An equality of rational divisors is coefficientwise for the chosen representatives. In particular, rD being an integral principal divisor means

for an actual rational function v; it includes integrality of every coefficient of rD.

For an integral Weil divisor G on a normal variety X, the reflexive divisorial sheaf is

For a rational divisor D, its rounded system is the complete system of \OO_X(\lfloor D\rfloor). A nonzero rational function v is a section exactly when

because its valuations are integers. This criterion does not require D or \lfloor D\rfloor to be Cartier. The field generated by a nonempty system is the subfield of \C(X) generated by all ratios of nonzero sections. It distinguishes the full embedded field from a proper subfield of the same transcendence degree.

A subset of \R satisfies the descending chain condition (DCC) if it contains no infinite strictly decreasing sequence. The ascending chain condition (ACC) is defined by reversing the inequalities. Both enter the proof: threshold ACC gives a DCC set for discriminant coefficients, and global ACC will reduce certain numerically trivial pairs to finite coefficient sets.

The index input needed for the relative theorem is in fibre dimension at most three. We use the following established theorem: for every finite set I\subset[0,1]\cap\Q, there is a positive integer n(I) such that every projective slc pair (T,B_T) of dimension at most three with coefficients in I and K_T+B_T\sim_\Q0 satisfies

This includes normal pairs and the zero boundary [16, Corollary 1.6, arXiv version 1]. We may enlarge n(I) to clear I. The conclusion means that the log pluricanonical sheaf \OO_T(n(I)(K_T+B_T)) is invertible and trivial on T itself. A choice of generator is a global form whose restrictions to the normalized components satisfy the conductor gluing. This global trivialization, rather than separate trivializations on the components, is the input required by the residue argument.

We also need this conclusion when the coefficients initially lie in a fixed rational DCC set I\subset[0,1]. On each normalized irreducible component, include the conductor with coefficient one. The resulting pair is projective lc with effective boundary coefficients in the DCC set I\cup\{1\}, and its adjoint is numerically trivial. Global ACC [15, Theorem 1.5] places all its nonzero coefficients in a fixed finite rational subset, depending only on I and the dimension bound. The coefficients of B_T therefore also lie in a fixed finite set. Applying (2.1) to the slc pair T supplies one uniform global trivialization, including the gluing. The possibly larger degree used later to restrict an ambient pluricanonical sheaf to T may depend on the ambient pair; it is kept separate from this uniform index.

3 An exact pullback and the coefficient to be controlled

We begin the proof of Theorem 1.1. Put k=\dim Z and K=\C(Z). The contraction condition makes K algebraically closed in \C(X), so this function-field extension is regular in characteristic zero. The generic fibre is therefore geometrically integral. It is normal by localization from X, and normality is geometric for a finite-type variety over the perfect field K. Choose a rational top form on Z. Together with the form defining K_X, the determinant sequence for function-field differentials determines a rational relative top form on the generic fibre X_\eta. We use its divisor as K_{X_\eta}. At codimension-one points of X_\eta, which are smooth over K, this divisor is the coefficientwise localization of K_X; vertical primes disappear. If d=k, then X_\eta=\Spec K and the relative form is a nonzero 0-form with zero divisor. Restrict a log resolution and its crepant sub-boundary to the geometric generic fibre. Generic smoothness makes the horizontal strata SNC there, so the original geometric generic pair is lc. That pair is projective of dimension d-k\leq3, its boundary coefficients belong to \Phi, and its adjoint is rationally linearly trivial.

The lower-dimensional index theorem applies to this pair over \overline K. For completeness, K is finitely generated over \C, so \overline K and \C are algebraically closed fields of characteristic zero with the same transcendence degree over \Q. An abstract field isomorphism therefore identifies the pair with a complex pair to which the theorem applies.

Choose a common integer p_0, also clearing \Phi, which principalizes the adjoint of every such geometric generic fibre. This principalization descends to K without enlarging the degree. Indeed, choose a finite Galois extension L/K over which a principalizing function is defined. The quotient of any two of its Galois conjugates has zero divisor on the proper geometrically integral normal fibre over L, hence belongs to L^*. These quotients form a multiplicative Galois cocycle. Hilbert 90 rescales the function to make it invariant, and it then descends to K. Consequently there is \psi\in\C(X)^* such that

is vertical over Z.

Choose a divisible by p_0 and \phi\in\C(X)^* with

The divisor of \psi^{a/p_0}/\phi is vertical. On the proper normal generic fibre this function has neither zeros nor poles, so it belongs to K^*. Denoting it by h, set

Substitution gives the first equality of (1.1) as an equality of actual rational divisors. In particular, D_Z is rational Cartier and D_Z\sim_\Q D.

We can now apply the lc-trivial-fibration theorem [11, Theorem 3.6]. Besides generic log canonicity and the rational pullback relation, it requires a rank-one condition. On a log resolution, the modified discrepancy divisor \mathbf A^* omits the discrepancy -1 terms. Because 0\leq B\leq1, rounding up leaves, over the generic point of the base, only an effective exceptional divisor over X: strict transforms contribute zero. The associated divisorial sheaf pushes forward to \OO_X by normality. Pushing on to Z thus has rank one, as required.

For a prime divisor P on any birational base model, let t_P be the lc threshold of its pullback over the generic point of P, computed with the crepant sub-pair on a resolved diagram. The discriminant coefficient is 1-t_P. The crepant sub-pair is sub-lc, so t_P\geq0; in the klt case it is sub-klt and t_P>0. These inequalities on every base model give the asserted generalized singularities. On the original base Z, the boundary upstairs is effective and a pullback component dominating P has positive integral multiplicity. Hence t_P\leq1, which gives B_Z\geq0.

For primes P on Z, the thresholds satisfy ACC uniformly. Shrink about the generic point of P so that P is Cartier. In a log resolution computing the threshold, only divisors with positive order along the pullback of P matter. Remove the images of those whose images are proper subsets of P. Then t_P is an ordinary lc threshold of an effective Cartier divisor, with positive integral coefficients, against an lc pair with coefficients in \Phi. The threshold ACC theorem [15, Theorem 1.1] applies. These thresholds are rational and lie in [0,1], so their complements form the required rational DCC set \mathcal B\subset[0,1].

The qualitative theorem makes the moduli b-divisor b-nef and gives its descent to a nef rational Cartier divisor on a sufficiently high smooth projective model W\to Z. This holds for our chosen presentation: changing a rational trivialization adds a rational principal b-divisor from the base, which preserves nefness and rational Cartier descent. Write

For P\subset W, put \alpha=\operatorname{coeff}_P D_W. Since \operatorname{coeff}_P B_W=1-t_P and the coefficient of K_W is an integer,

It remains to prove, uniformly in P, that

Then pM_W is an integral divisor on the smooth variety W, hence Cartier.

4 Reduction to a dlt fibre over a curve

Our objective is the integrality condition (3.1). Fix a prime P\subset W. Take a smooth projective model V\to X mapping to W by g, with SNC markings for the crepant boundary B_V^c, the exceptional divisors over X, and the support of g^*P. The morphism g has connected fibres: its function field extension is \C(W)=\C(Z)\subset\C(V)=\C(X), which is regular. Its Stein factor is therefore finite birational over the normal variety W and hence is W itself. Let \theta_V be the rational top form compatible with the chosen canonical divisor on X, and put K_V=\Div(\theta_V). Crepant pullback gives

Put t=t_P. At the generic point of P, the boundary B_V^c+t g^*P has coefficients at most one, with equality on at least one component dominating P. This is the SNC threshold calculation.

Choose sufficiently general very ample divisors A_1,\ldots,A_{k-1} on W and put C=A_1\cap\cdots\cap A_{k-1}, taking C=W when k=1. Bertini, applied successively to the pullback linear systems and the finitely many marked strata, makes C and V_C=g^{-1}C smooth, with SNC restricted markings. The curve C is connected, and the restriction of g has connected fibres, so V_C is connected and hence integral. We also require C\not\subset\Supp D_W and V_C\not\subset\Supp K_V\cup\Supp B_V^c\cup\Supp\Div(\psi). The dimension of V_C is d-k+1\leq4. Choose an intersection point c\in C\cap P transverse and general on P. We may avoid at c all other components of \Supp D_W, the bad strata loci on P, and the images of all relevant nondominating strata which do not contain P.

Let h:V_C\to C, F=h^*[c], and B^c=B_V^c|_{V_C}. We specify the canonical divisor in complete-intersection adjunction in order to retain the coefficient of P. Choose general representatives A_j'\sim A_j avoiding c, and rational functions \chi_j\in\C(W)^* with \Div(\chi_j)=A_j'-A_j. Define \theta_C as the iterated Poincaré residue, in the order of the cuts, of

along g^{-1}A_1,\ldots,g^{-1}A_{k-1}, taking each residue on the preceding cut. The generality of the divisors makes this a nonzero rational top form on V_C. For k=1 the product is empty and the residue is the identity. Each \chi_j supplies a simple pole along the cutting divisor and a zero along its chosen representative. Cancelling the cutting divisors before restriction, the residue formula gives an equality of actual divisors:

Set \psi_C=\psi|_{V_C} and D_C'=(D_W+\sum_jA_j')|_C. All these restrictions are defined by generality. Restricting (4.1) now gives

The coefficient assertion follows because C meets P transversely at c, while the A_j' and all other components of \Supp D_W avoid c.

Along F, the coefficients of B^c+tF are at most one, with equality on a component of F. Indeed, the marked divisors dominating P restrict transversely and reducedly, and near the selected point g^*P|_{V_C}=F. Over the generic point of C there is still a projective birational comparison with the normal fibre of the original model X, and the restricted exceptional divisors are exceptional there. To justify this last assertion, the generic point of C lies where W\to Z is an isomorphism. Generic flatness and openness of geometric normality and integrality give normal integral original fibres on an open base, and the birational isomorphism locus is fibrewise dense there. Shrinking further makes the exceptional images have fibre codimension at least two.

On V_C define an effective boundary \Gamma by taking F_{\mathrm{red}}, the horizontal strict transforms of the original boundary with their original coefficients, and all horizontal restricted exceptional divisors with coefficient one. Then (V_C,\Gamma) is log smooth, its horizontal coefficients belong to the fixed set \Phi\cup\{1\}, and

is effective near c and has coefficient zero on a component of F. On the generic fibre E is effective: its horizontal nonexceptional terms cancel, while every horizontal exceptional term has coefficient 1-\operatorname{coeff}B^c\geq0. Its support there is exceptional over the original normal fibre by the preceding birational comparison. Equation (4.2) gives the exact identity

We run a minimal-model program over C. All negative coefficients of E are vertical and occur over finitely many points, so there is an effective rational divisor A on C with E^+:=E+h^*A\geq0. The preceding identity becomes

In particular, K_{V_C}+\Gamma\sim_{\Q,C}E^+; its restriction to a very general fibre is rationally linearly equivalent to an effective divisor, so it is pseudo-effective over C.

The existence and preservation results recalled in [7, Remark 2.11, arXiv version 2] permit an MMP with scaling of an ample divisor over the base for an lc generalized pair whose underlying variety is \Q-factorial klt. We apply this to the \Q-factorial dlt pair (V_C,\Gamma) over C; the same reference states that its stages remain \Q-factorial and dlt. Each birational step is negative for the transformed adjoint. The zero nef datum satisfies the source’s NQC hypothesis. In dimension four, [7, Theorem 1.1, arXiv version 2] terminates every sequence of flips over C for an NQC lc generalized pair whose adjoint is pseudo-effective over C. The three-dimensional statement [7, Theorem 1.2, arXiv version 2] has no pseudo-effectivity hypothesis.

These results supply the program and terminate a possible flip tail. We check that its endpoint is nef and retains the exact identity. Use \theta_C to choose canonical divisors on every birational stage h_i:V_i\to C, and let \Gamma_i and E_i^+ denote the pushforwards. Every step is over C, so the preceding identity pushes forward to

Thus the adjoint remains pseudo-effective over C. A fibre-type negative extremal contraction is impossible: a general positive-dimensional contracted fibre has a curve \ell not contained in \Supp E_i^+. Since V_i is \Q-factorial, a Cartier multiple of E_i^+ restricts to an effective divisor on the normalization of \ell. Hence

a contradiction. Divisorial contractions strictly decrease the relative Picard number, so an infinite program would eventually consist only of flips. The cited termination statements exclude such a tail in dimensions three and four; in dimensions at most two no flips occur. The endpoint is a projective morphism f_N:N\to C with N \Q-factorial, (N,\Gamma_N) dlt, and K_N+\Gamma_N nef over C. Its unchanged regular function-field extension and Stein factorization give connected fibres, so f_N is a contraction.

Let E_N^+ be the transform of E^+ on N. The pushforward of h^*A is f_N^*A, so E_N:=E_N^+-f_N^*A is the pushforward of the original E. Substituting E_N^+=E_N+f_N^*A into the transformed identity gives

We claim that E_N=0 near c. Take a common resolution with maps r:\widetilde V\to V_C and r':\widetilde V\to N. Discrepancy comparison for the K+\Gamma MMP gives

by [19, Lemmas 3.38–3.39]. Pulling back the exact input and output identities cancels the common base pullbacks and the common principal divisor, giving

Let \eta be the generic point of C and q:\widetilde V_\eta\to X_\eta the morphism to the normal original fibre. The restriction of E_N to N_\eta is effective, being the pushforward of the effective restriction of E; hence the restriction of r'^*E_N is effective as well. The restriction of r^*E is effective and q-exceptional, so (4.5) makes the restriction of r'^*E_N q-exceptional. Finally, (4.4) gives E_N\equiv_C K_N+\Gamma_N, so that restriction is nef, in particular q-nef. The negativity lemma makes it zero. Thus E_N has no horizontal component. After shrinking about c, it is effective and supported on the special fibre, and it remains nef over C.

Write

and set \lambda=\max_i(q_i/e_i). The divisor \Delta=\lambda F_N-E_N is effective, anti-nef over C, and misses at least one component of the fibre. Suppose the fibre has positive dimension and \Delta\ne0. Connectedness gives a component S outside \Supp\Delta that meets it. Since N is \Q-factorial, choose a positive integer b so that b\Delta is Cartier. Its restriction to the integral projective component S is a nonzero effective Cartier divisor. If n=\dim S and A_S is a very ample Cartier divisor on S, then

Here A_S^{n-1} is represented by an effective curve cycle in the fibre (by S itself when n=1), so anti-nefness gives the opposite inequality. This contradiction proves \Delta=0, hence E_N=\lambda F_N. For a zero-dimensional fibre, f_N is an isomorphism of normal curves and the same proportionality is immediate. Finally,

and the original E has coefficient zero on a component of F. Taking the coefficient of its strict transform in (4.5) forces \lambda=0, whether or not that component survives on N. This proves the claim.

Shrink C about c and set T=(f_N^*[c])_{\mathrm{red}}. No divisor was extracted in the program, so every surviving component of this fibre has coefficient one in \Gamma_N. Thus \Gamma_N=T+H, where H is horizontal and has coefficients in the fixed set \Phi\cup\{1\}. The reduced fibre T is connected and projective. Its components have dimension d-k\leq3, since they are the components of the effective Cartier divisor f_N^*[c] on the integral variety N. Removing all boundary from the \Q-factorial dlt pair shows that N is klt. Shrink again so that D'_C is supported at most at c. For the integral Weil divisor L=p_0(K_N+T+H), the exact relation is now

5 The denominator detected by residues on the whole fibre

In the application to (4.6), the coefficient below is \beta=\alpha+t. Write p_0\beta=a/m in lowest terms, with m>0. An mth root of a base uniformizer produces a log pluricanonical generator with character \zeta^{-a} for \zeta\in\mu_m. We compare its p/p_0th power with the pullback of a degree-p form on the whole reduced fibre. Divisible adjunction will first make their ratios constant on each normalized component. Residues along the conductor will then make those constants equal, forcing the character to be trivial and hence m\mid p/p_0.

Apply Lemma 5.1 to (4.6), with \beta=\alpha+t and the fixed coefficient set \Phi\cup\{1\}. It gives a single multiple p of p_0 for which (3.1) holds at every prime P of the determination W. Consequently pM_W is integral and Cartier. Since the moduli b-divisor descends on W, the b-divisor p\mathbf M is b-Cartier. All constructions used only the dimensions at most four and the fixed coefficient set. This completes the proof of Theorem 1.1.

6 Effective systems recovering the entire base field

The exact presentation in the denominator theorem identifies complete section spaces on the total space and the base. We combine this identification with effective birationality on a base model.

7 Integral principal multiples over rationally connected bases

We now bound a common principal multiple of the actual generalized adjoint, including its denominators. This will supply uniform trivializations on rationally connected bases.