Nash problem for a toric pair and the minimal log-discrepancy

TL;DR

This paper addresses the Nash problem for toric pairs, proving it affirmative, and shows how minimal log-discrepancy relates to Nash components, including cases with infinite values.

Contribution

It formulates the Nash problem for toric pairs with invariant ideals and proves the minimal log-discrepancy is computed by Nash divisors when finite.

Findings

Nash problem is affirmatively solved for toric pairs.

Minimal log-discrepancy is computed by Nash divisors when finite.

Existence of Nash components with negative log-discrepancy when minimal log-discrepancy is -/infty.

Abstract

This paper formulates the Nash problem for a pair consisting of a toric variety and an invariant ideal and gives an affirmative answer to the problem. We also prove that the minimal log-discrepacy is computed by a divisor corresponding to a Nash component, if the minimal log-discrepancy is finite. On the other hand there exists a Nash component such that the corresponding divisor has negative log-discrepancy, if the minimal log-discrepancy is $-/infty$.