# On the Nash problem for terminal threefolds of type $cA/r$

## TL;DR

This paper investigates Nash and essential valuations of terminal threefolds of type cA/r, providing complete descriptions in certain cases and constructing counterexamples in others.

## Contribution

It offers a comprehensive analysis of Nash valuations for cA/r threefolds, including new counterexamples for the Nash problem in non-Gorenstein or non--factorial cases.

## Key findings

- Complete description of valuations when r=1 or threefold is -factorial
- Construction of counterexamples for non-Gorenstein or non--factorial cases
- Identification of conditions where Nash valuations can be fully characterized

## Abstract

We study Nash valuations and essential valuations of terminal threefolds of type $cA/r$. If $r=1$ or the given threefold is $\mathbb Q$-factorial, then all the Nash valuations and essential valuations can be completely described. We construct non-Gorenstein or non-$\mathbb Q$-factorial counter examples for the Nash problem.