The purpose of this paper is to demonstrate the power of geometrical methods which allow one to substantially improve the log minimal model program (LMMP) for 3-folds. This means that the basic facts such as the cone, contraction, flip, termination, and abundance theorems hold under much more general and, perhaps, ultimate conditions: 9 boundaries are R-divisors and 9 singularities are log canonical.Of course, we do not pretend to give a direct proof of them, except for the termination theorem. It is only a derivation of these facts from the well-known situation where the boundaries are Q-divisors and singularities are log terminal [11, 15]. In the latter case, the proofs use essentially cohomological methods: the vanishing and nonvanishJng theorems.The improvement pursues the standard mathematical goal of being close to the cutting edge, but it gives some new applications as well. The most important of them are the two main theorems in Sec. 6, which describe the behavior of log models with respect to their boundaries. In particular, the second of them answers in the affirmative regarding [24, Problem 6] in dimension 3 and improves substantially the first attempt [25, Relative Model Theorem].Another application presents results on the Kleiman-Mori cone in the critical zone where the log canonical divisor is trivial.As we see in Sec. 6, the core of the methods is based on the standard LMMP with Q-boundaries and log terminal singularities and it works in any dimension, except for the log termination, which is given for 3-folds in Sec. 5. The log termination for 3-folds in the standard case is credited to Kawamata [10].Other sections contain preparatory and related materials.Acknowledgment. Some gaps in the first draft were pointed out by T. Hayakawa and S. Mori in Theorem 3.2, as well by A. Bruno and K. Matsuki in the contraction results (6.16 and 6.16.1). The author would like to express thanks to all of them.

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To construct a resulting model in LMMP, it is sufficient to prove existence of log flips and their termination for certain sequences. We prove that LMMP in dimension d − 1 and termination of terminal log flips in dimension d imply, for any log pair of dimension d, existence of a resulting log model: a strictly log minimal model or a strictly log terminal Mori log fibration, and imply existence of log flips in dimension d + 1. As consequence, we prove existence of a resulting model of 4-fold log pairs, existence of log flips in dimension 5, and Geography of log models in dimension 4.