On uniformly rational varieties

TL;DR

This paper explores the properties of uniformly rational varieties, investigates criteria distinguishing them from all smooth rational varieties, and proves their uniform rationality in specific cases involving resolutions of nodal cubic threefolds.

Contribution

It introduces potential criteria for identifying uniformly rational varieties and proves their uniform rationality for certain resolutions of nodal cubic threefolds.

Findings

Small algebraic resolutions of nodal cubic threefolds are uniformly rational.

Big resolutions of nodal cubic threefolds are uniformly rational.

Discussion of criteria distinguishing uniformly rational varieties from all smooth rational varieties.

Abstract

We investigate basic properties of uniformly rational varieties, i.e. those smooth varieties for which every point has a Zariski open neighborhood isomorphic to an open subset of A^n. It is an open question of Gromov whether all smooth rational varieties are uniformly rational. We discuss some potential criteria that might allow one to show that they form a proper subclass in the class of all smooth rational varieties. Finally we prove that small algebraic resolutions and big resolutions of nodal cubic threefolds are uniformly rational.