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

TL;DR

This work investigates the Nash problem for 3-fold terminal singularities, with a focus on type cAx/2. It proves that exceptional divisors computing the minimal discrepancy of these singularities (specifically $a(E,X)=1/2$) generate Nash valuations, aligning with toric intuition and supporting conjectures that link low discrepancies to Nash valuations in the general 3-fold terminal setting. The authors develop a framework using discrepancies, divisorial contractions, and weighted blow-ups to propagate Nash valuations from minimal-discrepancy divisors and provide partial evidence in the Gorenstein case through explicit constructions and examples. Together, the results suggest a broader principle: low-discrepancy exceptional divisors are intimately connected to the Nash valuation image, guiding future classification and resolution strategies for terminal singularities.

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.

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2: IK03
• Conjecture 1.3: Conjecture A
• Conjecture 1.4: Conjecture B
• Theorem 1.5
• Theorem 2.1: Reid83
• Theorem 2.2: Dan83MS84Mori85Hay99Hay00Chen16
• Theorem 2.3: KSB88
• Theorem 2.4: Kaw93Mar96
• Definition 2.5: Hay99 and Hay00