On the crepant blowing-ups of canonical singularities and its application to degenerations of surfaces (Q1084142)

This note gives a sketch of the proof of the following theorem: \(Let\quad X\) be a 3-dimensional normal complex algebraic variety with at most canonical singularities. Then the ring \(\oplus_{m\geq 0}{\mathcal O}_ X(m.D)\quad is\) finitely generated, where D is a Weil divisor on X. This result is then applied to prove the existence of minimal models for on-parameter-families of surfaces of non-negative Kodaira-dimension whose degenerate members are reduced and have only simple normal crossings.