The motivic nearby fiber and degeneration of stable rationality

TL;DR

This paper demonstrates that stable rationality is preserved under certain degenerations using motivic techniques, providing new tools to study rationality problems in algebraic geometry.

Contribution

It introduces a motivic specialization approach to prove the stability of rationality in families with specific singularities, advancing the understanding of rationality degeneration.

Findings

Stable rationality specializes in families with ordinary double point singularities.

Motivic specialization techniques are effective in rationality problems.

Provides criteria for stable rationality in the Grothendieck ring of varieties.

Abstract

We prove that stable rationality specializes in regular families whose fibers are integral and have at most ordinary double points as singularities. Our proof is based on motivic specialization techniques and the criterion of Larsen and Lunts for stable rationality in the Grothendieck ring of varieties.