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

arXiv comment5 pages. v2: Fix a serious mistake pointed out by the referee. v3: The argument has a serious gap, as pointed out by Colliot-Thélène. Now the theorem holds over Laurent fields in one variable. A result about R-equivalence on cubic surfaces is added in this version. Title changed. Comments are welcome

References in corpus (1)

• Weak Approximation for Isotrivial Families

Cited by in corpus (1)

• Weak Approximation for Cubic Hypersurfaces and Degree 4 del Pezzo Surfaces