On nonrational divisors over non-Gorenstein terminal singularities

Let (X, o) be a germ of a 3-dimensional terminal singularity of index m ≥ 2. If (X, o) has type cAx/4, cD/3-3, cD/2-2, or cE/2, then we assume that the standard equation of X in ℂ4/ℤ m is nondegenerate with respect to its Newton diagram. Let π: Y → X be a resolution. We show that there are at most 2 nonrational divisors E i , i = 1, 2, on Y such that π(E i ) = o and the discrepancy a(E i , X) is at most 1. When such divisors exist, we describe them as exceptional divisors of certain blowups of (X, o) and study their birational type. © Springer Science+Business Media, Inc. 2007.