On the Nash Problem over 3-Fold Terminal Singularities

Abstract

In this dissertation, we study the Nash problem over terminal singularities in dimension 3, or 3-fold terminal singularities. In particular, for 3-fold terminal singularities, we propose two conjectures that aim to characterize their induced Nash valuations. (A) Any prime divisor with discrepancy bounded by 1 induces a Nash valuation. (B) Any prime divisor with minimal discrepancy induces a Nash valuation. We prove that Conjecture A holds for toric terminal singularities in arbitrary dimension, and Conjecture B holds for 3-fold terminal singularities of type cAx/2. In the Gorenstein cases, we provide a partial result with examples for both of the conjectures.