Retract rational varieties are uniformly retract rational
Juliusz Banecki
TL;DR
The paper proves that every nonsingular retract rational variety $X$ over an infinite field $\mathbb{K}$ is uniformly retract rational, establishing $X$ as locally biregular to a Zariski open subset of $\mathbb{K}^n$. This is achieved through a suite of technical lemmas: a lifting proposition that converts a rational map into a regular germ via an ambient perturbation, a Noether normalization/transversality framework ensuring finiteness and control of associated primes, and a deformation-analytic argument to preserve regularity under perturbations. As a consequence, every nonsingular rational complex projective variety is algebraically elliptic, linking retract rationality with algebraic ellipticity. The results advance the understanding of the relationship between retract rationality, uniform retract rationality, and ellipticity, with implications for the geometry of complex projective varieties.
Abstract
We prove that nonsingular retract rational algebraic varieties over any infinite field are uniformly retract rational. As a consequence, every rational, projective, nonsingular complex variety is algebraically elliptic.