Effective log Iitaka fibrations for surfaces and threefolds

TL;DR

This paper establishes bounds for log Iitaka fibrations on surfaces and threefolds, providing new insights into the structure of klt pairs and their fibrations, with implications for boundedness in algebraic geometry.

Contribution

It proves an analogue of the 'bounding the denominators' result for log canonical bundles and demonstrates the existence of a universal N for Iitaka fibrations of certain klt pairs.

Findings

Bound N depends only on the coefficients set for the Iitaka fibration.

Existence of a universal N for klt pairs of Kodaira codimension one in dimension up to three.

Birational boundedness for klt surfaces of general type.

Abstract

We prove an analogue of Fujino and Mori's ``bounding the denominators'' in the log canonical bundle formula (see also Prokhorov and Shokurov) for Kawamata log terminal pairs of relative dimension one. As an application we prove that for a klt pair $(X,\Delta)$ of Kodaira codimension one and dimension at most three such that the coefficients of $\Delta$ are in a DCC set $\mathcal{A}$, there is a natural number $N$ that depends only on $\mathcal{A}$ for which the round down of $\N(K_X+\Delta)$ induces the Iitaka fibration. We also prove a birational boundedness result for klt surfaces of general type.