Minimal Discrepancy for a Terminal cDV Singularity Is 1

Abstract

An answer to a question raised by Shokurov on the minimal discrepancy of a terminal singularity of index 1 is given. It is proved that the minimal discrepancy is 1 (it is 2 for a non-singular point and 0 for all other canonical singularities of index 1). A rough classification of terminal singularities of index 1 based on finding certain low degree monomials in their equations, and the toric techniques of weighted blow ups are used. This result has been generalized to terminal singularities of index $r>1$ by Y.Kawamata; his theorem states that the minimal discrepancy is $1/r$