\(R\)-equivalence on low degree complete intersections (Q2913221)

Let \(k\) be a field of characteristic zero. Let \(X\) be a projective geometrically integral variety over \(k\) with a fixed ample divisor, and let \(\bar{M}_{0,2}(X,d)\) denote the Kontsevich moduli space for all genus zero stable curves over \(X\) of degree \(d\) with two marked points. We say that \(X\) is \(k\)-rationally simply connected if \(H_2(X,\mathbb{Z})\) has rank one and for any sufficiently large integer \(e\) there exists a geometrically irreducible component \(M_{e,2}\subset\bar{M}_{0,2}(X,e)\) intersecting the open locus of irreducible curves \(M_{0,2}(X,e)\) and such that the restriction of the evaluation morphism \(M_{e,2}\rightarrow X\times X\) is dominant with rationally connected general fiber. In this paper the author proves that if \(k\) is either a function field in one variable over \(\mathbb{C}\) or the field \(\mathbb{C}((t))\) and \(X\) is a \(k\)-rationally simply connected variety over \(k\), then \(R\)-equivalence on rational points of \(X\) is trivial and that the Chow group of zero-cycles of degree zero is trivial too. In particular, this holds for a smooth complete intersection of \(r\) hypersurfaces in \(\mathbb{P}_k^n\) of respective degrees \(d_1,\dots,d_r\) with \(\sum_{i=1}^rd_i^2\leq n+1\).