On the Nash Problem over 3-Fold Terminal Singularities of Type cAx/2

TL;DR

This paper investigates Nash valuations on 3-fold terminal singularities, focusing on type cAx/2, and proposes a conjecture that minimal discrepancy divisors induce Nash valuations across all such singularities.

Contribution

It establishes that in cAx/2 singularities, divisors computing minimal discrepancy induce Nash valuations and conjectures this holds generally for all 3-fold terminal singularities.

Findings

Exceptional divisors in cAx/2 induce Nash valuations.

Minimal discrepancy divisors are linked to Nash valuations.

Evidence supports the conjecture in Gorenstein cases.

Abstract

We study Nash valuations on 3-fold terminal singularities, especially in type cAx/2. We find that, in type cAx/2, exceptional prime divisors computing the minimal discrepancy (which is 1/2 in this case) induce Nash valuations. We conjecture this in general for all 3-fold terminal singularities, and provide some evidence in the Gorenstein case.