Effective log Iitaka fibrations for fourfolds

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Effective log Iitaka fibrations for fourfolds

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Take any log canonical fourfold pair whose boundary coefficients come from a fixed finite set Φ. This class shows you how to work out which theorem supplies the single degree m(Φ) that makes ⌊mD⌋ nonempty and makes its section ratios generate the whole Iitaka field. It also shows why the hardest case comes down to a uniform canonical index N₄ for klt fourfolds with K_X ∼_Q 0.

The route to m(Φ)

Asking for the right image dimension is not enough. The ratios could generate a smaller field, with K(D) only a finite extension of it [0:41].
Step one: get a semiample model. Pseudo-effectivity gives nonvanishing in some degree, but that degree is not uniform. A crepant dlt modification and an adjoint-negative program then reach a model (X′, Δ′) with D′ ∼_Q f*A. This model keeps every rounded section space V_m(D) in every degree, so K(D) = ℂ(Z) [0:53].
Step two: read dim Z. If dim Z ≥ 1, the relative big-base theorem gives the degree. If Z is a point, you have an index problem with three branches: not klt (Jiang–Liu), klt with Δ ≠ 0 (Xu), and klt with Δ = 0 (N₄). A common multiple of these degrees gives m(Φ) [1:34].

Practice case: (X, 0) klt with K_X ∼_Q 0. The base is a point and the boundary is zero, so only N₄ applies [2:21].

Why N₄ exists

The reductions leave a terminal V with irregularity zero. Now assume r → ∞ [2:48]. On the cyclic cover, the Seshadri constant grows like r^{1/4}. On the diagonal product Z_t = Y^t/μ_r, though, curve cost is bounded by C·t², with no dependence on r. That bound is proved in characteristic p by comparing jets with Frobenius [3:24]. The tension gives a degree-one forgetting map, then a rational vector field, and its flow produces uncountably many birational self-maps [4:17]. But q = 0 and Matsumura's theorem make Bir(V) countable. So r is bounded [4:47].

Rule to take with you: pass to a semiample model that keeps every section space, read the dimension of the base, and let it decide which uniform input you need. One question stays open: can a finite Φ be replaced by a DCC set?

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