Prelimiting Flips

The paper discusses an inductive approach to constructing log flips. In addition to special termination and thresholds, we introduce two new ingredients: the saturation of linear systems, and families of divisors with confined singularities. We state conjectures concerning these notions in any dimension and prove them in general in dimension $\le 2$. This allows us to construct prelimiting flips (pl flips) and all log flips in dimension 4 and to prove the stabilization of an asymptotically saturated system of birationally free (b-free) divisors under certain conditions in dimension 3. In dimension 3, this stabilization upgrades pl flips to directed quasiflips. It also gives for the first time a proof of the existence of log flips that is algebraic in nature, that is, via f.g. algebras, as opposed to geometric flips. It accounts for all the currently known flips and flops, except possibly for flips arising from geometric invariant theory.