On a Computational Approach to the Nash Blowup Problem

TL;DR

The paper develops and implements a computational framework to study Nash blowups and normalized Nash blowups for toric varieties, translating geometric questions into combinatorial data on cones and affine semigroups. Using a SageMath-based pipeline, the authors uncover explicit counterexamples to the Nash blowup conjecture in dimension $3$ and higher and to the normalized Nash blowup conjecture in dimensions $4$ and $5$, while also reporting extensive positive results in low dimensions (e.g., $2$D non-normalized and $3$D normalized cases). They construct illustrative instances of singularities (hypersurfaces, cyclic quotients, and $\mathbf{Q}$-Gorenstein types) that resist resolution by iterated normalized Nash blowups and map out large-scale computational evidence across millions of toric varieties. The work provides both definitive counterexamples and substantial positive data, guiding future inquiries into invariants and subdivision schemes that might govern resolution behavior in the toric setting.

Abstract

In this paper we describe the implementation that led to the counterexamples to the Nash blowup conjectures recently discovered by the authors. We also provide new examples of toric varieties with prescribed singularities that are not resolved by the normalized Nash blowup, including cyclic quotient singularities, toric hypersurfaces, and Q-factorial Gorenstein singularities. In addition, we report extensive computational evidence: tens of thousands of two-dimensional toric varieties that are resolved by iterating the Nash blowup, and millions of three-dimensional toric varieties that are resolved by iterating the normalized Nash blowup. This provides positive evidence for the remaining open cases of the conjectures.

Figures (5)

• Figure 2: The resulting subdivision of the original cone $\sigma$.
• Figure : Cone $\sigma$.
• Figure : Cone $\sigma$.
• Figure : Cone $\sigma^\vee$.
• Figure : Hilbert Basis.

Theorems & Definitions (38)

• Example 2
• Example 3
• Definition 4
• Remark 5
• Definition 6
• Definition 7
• Definition 8
• Remark 9
• Example 10
• Definition 11: Nash blowup