Decomposition of log crepant birational morphisms between log terminal surfaces (Q5933880)

The proper birational morphism \(h: (X_1,D_1) \rightarrow (X_2,D_2)\) between log terminal surfaces is ``log crepant'' if \(K_{X_1} + D_1 = h^*(K_{X_2} + D_2)\). If moreover \((X_1,D_1)\) and \((X_2,D_2)\) are Kawamata log terminal then it is known that \(h\) is a composite of log flopping type divisorial contraction morphisms over \(X_2\). In this paper, it is found a similar decomposition of \(h\) in the general case. The main theorems 1 and 2 show that every log crepant birational morphism between log terminal surfaces can be decomposed into log flopping type divisorial contraction morphisms and log blow-downs. Moreover, by repeating these two kinds of contractions one reaches a minimal log minimal surface starting from any log minimal surface.