Stable Limits of Log Surfaces and Cohen–Macaulay Singularities

Given a family of surfaces of general type over a smooth curve, one can apply semistable reduction and the minimal model program to obtain a stable reduction. This is the basis for a geometric compactification for moduli spaces of surfaces of general type, due to Kollár, Shepherd-Barron, and Alexeev. However, this approach hinges on the fact that the resulting stable limit has relatively mild singularities; in particular, it should be Cohen–Macaulay. Unfortunately, the standard formalism does not guarantee that stable limits of families of log surfaces are Cohen–Macaulay. Here we prove that this is the case.