Two two-dimensional terminations

Varieties with log terminal and log canonical singularities are considered in the Minimal Model Program, see {...} for introduction. In {shokurov:hyp} 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. 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. In this paper we prove that the following two sets satisfy the ascending chain condition: 1. The set of minimal log discrepancies for KX+B where X is a surface with log canonical singularities. 2. The set of groups (b₁,...bₛ) such that there is a surface X with log canonical and numerically trivial KX+∑ bⱼBⱼ. The order on such groups is defined in a natural way.