On rational singularities and counting points of schemes over finite rings

TL;DR

This paper explores the relationship between rational singularities of schemes over integers and the asymptotic behavior of point counts over finite rings, establishing equivalences under certain geometric conditions.

Contribution

It proves that boundedness of normalized point counts over finite rings characterizes schemes with rational singularities, completing previous work by Aizenbud and Avni.

Findings

Bounded normalized point counts imply rational singularities.

Reducedness of the generic fiber is equivalent to certain point count bounds.

The work completes a prior result by Aizenbud and Avni.

Abstract

We study the connection between the singularities of a finite type $\mathbb{Z}$-scheme X and the asymptotic point count of X over various finite rings. In particular, if the generic fiber $X_{\mathbb{Q}}=X\times_{\mathrm{Spec}\mathbb{Z}}\mathrm{Spec}\mathbb{Q}$ is a local complete intersection, we show that the boundedness of $\frac{\left|X(\mathbb{Z}/p^{n}\mathbb{Z})\right|}{p^{n\mathrm{dim}X_{\mathbb{Q}}}}$ in p and n is in fact equivalent to the condition that $X_{\mathbb{Q}}$ is reduced and has rational singularities. This paper completes a result of Aizenbud and Avni.