R-equivalence on low degree complete intersections

TL;DR

This paper proves that for certain smooth complete intersections over specific fields, the R-equivalence relation among rational points is trivial, and the zero-cycle Chow group is zero, extending understanding of rational points and zero-cycles.

Contribution

It establishes conditions under which R-equivalence is trivial and zero-cycle Chow groups vanish for low degree complete intersections over particular fields.

Findings

R-equivalence on rational points is trivial under given conditions.

Chow group of zero-cycles of degree zero is zero for these intersections.

Results apply to fields of complex curves and Laurent series.

Abstract

Let $k$ be the function field of a complex curve or the field $C((t))$. We show that for a smooth complete intersection $X$ of $r$ hypersurfaces in $P^n_k$ of respective degrees $d_1,...,d_r$ with $\sum d_i^2\leq n+1$ the R-equivalence on rational points of $X$ is trivial and the Chow group of zero-cycles of degree zero $A_0(X)$ is zero.