R-equivalence on del Pezzo surfaces of degree 4 and cubic surfaces

TL;DR

This paper establishes the uniqueness of R-equivalence classes on certain algebraic surfaces over Laurent fields, specifically del Pezzo surfaces of degree 4 and some cubic surfaces, under specific geometric conditions.

Contribution

It proves the uniqueness of R-equivalence classes on del Pezzo surfaces of degree 4 over Laurent fields and on cubic surfaces with stable GIT conditions over complex Laurent series.

Findings

Unique R-equivalence class on del Pezzo degree 4 surfaces over Laurent fields.

Unique R-equivalence class on cubic surfaces with stable GIT conditions.

Results hold in characteristic not 2 or 5 for del Pezzo surfaces.

Abstract

We prove that there is a unique $R$-equivalence class on every del Pezzo surface of degree $4$ defined over the Laurent field $K=k((t))$ in one variable over an algebraically closed field $k$ of characteristic not equal to $2$ or $5$. We also prove that given a smooth cubic surface defined over $\mathbb{C}((t))$, if the induced morphism to the GIT compactification of smooth cubic surfaces lies in the stable locus (possibly after a base change), then there is a unique $R$-equivalence class.