Uniform effective log Iitaka fibrations for fourfolds

1. Separate product and rationally connected cases using the cover decomposition

The first reduction uses a decomposition on a finite quasi-étale cover and characters on volume forms. After separating product and abelian cases, the remaining single nonflat four-dimensional factor has the property that its positive-dimensional proper rational images are rationally connected (Ufour, Proposition 4.1–Lemma 4.2, pp. 10–14). In the noncanonical case the canonical divisor on a resolution is not pseudo-effective. Uniruledness, the MRC quotient, and the restriction on rational images then imply rational connectedness; the passage from non-pseudo-effectivity to uniruledness follows Corollary 4.4 (p. 15).

Proposition 4.1–Corollary 4.4 · pp. 10–15

2. Use relative denominators when an intermediate divisor exists

In the canonical case a crepant terminalization preserves the index. If an effective Weil divisor $A$ has $1\leq\kappa(A)\leq3$, use a semiample model for $(V,\delta A)$ with $\delta>0$ small. The proof checks crepancy for the boundary-free canonical divisor separately along this MMP. Pushing forward $rK_V=\operatorname{Div}(h)$ preserves the index, and the relative denominator theorem is applied to $(V',0)$, not to the varying coefficient $\delta$. On the rationally connected base, RD, Proposition 7.1, p. 17 gives an integral principal multiple of the generalized adjoint. The denominator of $\delta$ therefore does not enter the constant (Ufour, Lemma 4.6, pp. 15–16).

Lemmas 4.5–4.6 · pp. 15–16

3. Make residual equivariant fibres general type

For the remaining terminal fourfold and its canonical cyclic cover, the relevant intermediate equivariant rational fibrations have general-type smooth geometric generic fibres. Lemma 4.7 (pp. 16–17) uses bigness of a divisor pulled back from a descended base and positive terminal discrepancies to make the canonical divisor of the fibre big. This is the hypothesis needed for the next uniform estimates.

Lemma 4.7 · pp. 16–17

1. Apply scalar bounds using the general-type condition

Let $\gamma(P;V)$ be the supremum of section order divided by degree at a general point, using sufficiently divisible degrees. The scalar bounds of Theorem 6.1 (p. 23) assume the preceding general-type condition on equivariant fibrations. Applied to a normalized polarization $1\leq L^4\leq2$ and the canonical cyclic cover $\pi:Y\to V$ of index $r$, they give $\varepsilon(L)\geq c$ and $\varepsilon(\pi^*L)\geq cr^{1/4}$.

Definition 5.3・Theorem 6.1 · pp. 21, 23

2. Turn diagonal rank loss into contradictory vanishing orders

In contrast, Proposition 7.1 (pp. 33–42) bounds $\varepsilon(P_t)\leq Ct^2$ for the external-sum polarization $P_t$ on $Z_t=Y^t/\mu_{r,\mathrm{diag}}$. In characteristic $p$, a surplus of ordinary jets produces a matrix of full rank at a general point, but of rank at most $R/4$ on the Frobenius diagonal. A nonzero maximal minor must vanish there to order at least $3R/4$; slotwise scalar bounds and Lemma 7.4 give a strictly smaller order. LA §9 is a methodological predecessor here; its abundance conclusion is not an input to this jet calculation.

Proposition 7.1・Lemma 7.4 · pp. 33–42

3. Obtain degree-one forgetting from chain leaves

Proposition 8.1 (pp. 42–46) applies the two Seshadri estimates to chain leaves. Fix finitely many products $2\leq t\leq T$ before increasing the covering order $r$. Each leaf then maps generically finitely to any single slot, so its dimension lies between $1$ and $4$. Leaf degrees are bounded by $KT^8$. If every forgetting map along a constant-dimensional stretch had degree at least two, the degrees would grow exponentially. Some forgetting map must consequently have degree one.

Proposition 8.1 · pp. 42–45

4. Extend local flows birationally and bound the index

Degree one is essential: it converts evaluation of the missing coordinate from a finite correspondence into a rational map. Tangent transport and local flows then produce birational self-maps. A positive-dimensional flow gives uncountably many such maps, contradicting Lemma 8.2 (pp. 46–47), which proves countability of $\operatorname{Bir}(V)$ using irregularity zero on a smooth model. Section 9 (pp. 47–48) excludes unbounded index sequences and takes a common multiple of the bounded orders in all cases.

Proposition 8.1・Lemma 8.2・§9 · pp. 45–48

1. Construct a semiample model after qualitative nonvanishing

Initial nonvanishing comes from good models and semiampleness in LA, Theorems 11.1, 1.1. The degree of the resulting nonzero section depends on the input and is not yet uniform (Ufour, §10, p. 48). Proposition 2.2 (pp. 6–7) next uses a crepant dlt modification, termination of fourfold flips, and abundance after nonvanishing from SL, Theorem 1.2, p. 2 to obtain a model $(X',\Delta')$ with coefficient set $\Phi'=\Phi\cup\{1\}$.

Proposition 2.2・§10 · pp. 6–7, 48

2. Identify rounded section spaces in every integer degree

The comparison in every integer degree is

This is equation (2.1), checked through the pole inequality $\operatorname{Div}(v)+kD\geq0$ in (2.2) and an effective exceptional difference on a common resolution. A comparison only in selected Cartier degrees would not suffice. For the semiample contraction $f:X'\to Z$, one has $D'\sim_{\mathbb Q}f^*A$ with $A$ ample. Relative algebraic closedness for the contraction places every section ratio in $\mathbb C(Z)$.

(2.1)–(2.2)・Lemma 2.3 · pp. 6–8

3. Use effective systems with fixed denominators on a positive-dimensional base

When $\dim Z>0$, RD, Theorem 1.1, p. 2 gives an exact canonical bundle formula with fixed denominators, and RD, Proposition 6.1, pp. 15–17 recovers the entire base field in every positive multiple of a uniform degree. These appear as Ufour Theorem 3.1 and Proposition 3.2 (p. 8). The birational contraction case $\dim Z=4$ is included.

Theorem 3.1・Proposition 3.2・§10 · pp. 8, 48–49

4. Separate three cases over a point base

When $Z$ is a point, $D'\sim_{\mathbb Q}0$. The index is bounded by JL, Corollary 1.7, p. 2 in the non-klt case, by Xu, Theorem 1.14, p. 3 when the pair is klt with $\Delta'\neq0$, and by the present Theorem 1.1 when it is klt with $\Delta'=0$. Xu's hyperstandard convention includes the given finite set by using $\{1-b:b\in\Phi'\}$ and denominator one. Take a common multiple of these degrees and the positive-base degree; the latter permits this because its conclusion holds for every positive multiple. Equation (2.1) returns to the original complete systems.

§10 · p. 49

5. Preserve section ratios under a change of canonical representative

Replacing the canonical divisor by $K_X+\operatorname{Div}(h)$ changes the floor divisor by $m\operatorname{Div}(h)$. Multiplication of sections by $h^{-m}$ identifies the systems without changing their ratios, so the uniform degree is independent of the representative (Ufour, p. 49).

§10 · p. 49