On the Log Discrepancies in Toric Mori Contractions
  It was conjectured by McKernan and Shokurov that for all Mori contractions from X to Y of given dimensions, for any positive epsilon there is a positive delta, such that if X is epsilon-log terminal, then Y is delta-log terminal. We prove this conjecture in the toric case and discuss the dependence of delta on epsilon, which seems mysterious. 
Valery Alexeev and Alexander Borisov
math.AG