Two Two-dimensional Terminations

Varieties with log terminal and log canonical singularities are considered in the Minimal Model Program, see [KMM] for introduction. In [SH2] it was conjectured that many of the interesting sets, associated with these varieties have something in common: they satisfy the ascending chain condition, which means that every increasing chain of elements terminates (in [SH2] it was called the upper semi-discontinuaty). Philosophically, this is the reason why two main hypotheses in the Minimal Model Program: existence and termination of flips should be true and are possible to prove. As for the latter, one of the main properties of flips is that log discrepancies after doing one do not decrease and some of them actually increase, [SH1]. Therefore, if one could show that a set of “the minimal discrepancies” satisfies the ascending chain condition, that would help to prove the termination of flips. The Shokurov’s proof of existence of 3-fold log flips [SH3] is another example of applying the same principle. In fact, to complete the induction it uses some 1 dimensional statement, 2 dimensional analog of which is proved in this paper. For further discussion, see also [A-K]. For one of the first examples where the phenomenon is actually proved let us mention the following