R-equivalence on low degree complete intersections

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.