On the arithmetic of weighted complete intersections of low degree

TL;DR

This paper proves that smooth, low-degree weighted complete intersections are rationally simply connected, leading to significant arithmetic consequences such as weak approximation and trivial R-equivalence over function fields.

Contribution

It establishes rational simple connectedness for weighted complete intersections of low degree, a property not previously confirmed for this class of varieties.

Findings

Weighted complete intersections of low degree are rationally simply connected.

These varieties satisfy weak approximation over function fields.

R-equivalence of rational points is trivial on these varieties.

Abstract

A variety is rationally connected if two general points can be joined by a rational curve. A higher version of this notion is rational simple connectedness, which requires suitable spaces of rational curves through two points to be rationally connected themselves. We prove that smooth, complex, weighted complete intersections of low enough degree are rationally simply connected. This result has strong arithmetic implications for weighted complete intersections defined over the function field of a smooth, complex curve. Namely, it implies that these varieties satisfy weak approximation at all places, that R-equivalence of rational points is trivial, and that the Chow group of zero cycles of degree zero is zero.