Nash blowups of 2-generic determinantal varieties in positive characteristic

TL;DR

The paper addresses Nash blowups in positive characteristic for $2$-generic determinantal varieties, focusing on the case $M_{m,n}^2$. It achieves this via a two-step strategy: (i) provide an explicit toric realization $M_{m,n}^2 = X_{\Gamma}$ with a detailed semigroup $\Gamma$, establishing normality; (ii) develop a characteristic-free combinatorial description of Nash blowups for toric varieties using modulo-$p$ logarithmic Jacobian ideals $\mathcal{J}_p$ and the González–Teissier framework. The main result shows that the Nash blowup of $M_{m,n}^2$ is non-singular in positive characteristic, extending the characteristic-zero result of Ebeling–Gusein-Zade and holding without the need for normalization. This work broadens the positive-characteristic theory of Nash blowups for toric and determinantal varieties and provides explicit, characteristic-independent tools for studying their singularities and resolutions.

Abstract

We show that the Nash blowup of 2-generic determinantal varieties over fields of positive characteristic is non-singular. We prove this in two steps. Firstly, we explicitly describe the toric structure of such varieties. Secondly, we show that in this case the combinatorics of Nash blowups are free of characteristic. The result then follows from the analogous result in characteristic zero proved by W. Ebeling and S. M. Gusein-Zade.

Theorems & Definitions (33)

• Theorem 1: see Theorem \ref{['M2 toric']}
• Theorem 2: see Theorem \ref{['Nash M2 smooth']}
• Definition 1.1
• Proposition 1.2
• Definition 1.3
• Proposition 1.4
• Example 1.5
• Example 1.6
• Theorem 1.7
• proof