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.

中文翻译：

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低度加权完全交的算法

如果两个一般点可以通过有理曲线连接，则一个变体是有理连接的。这个概念的更高版本是有理简单连通性，它需要通过两点的有理曲线的合适空间才能有理地连接自己。我们证明了足够低度的平滑、复杂、加权的完全交集是合理简单连接的。该结果对于在平滑、复杂曲线的函数域上定义的加权完全交集具有很强的算术意义。也就是说，这意味着这些变体在所有地方都满足弱近似，有理点的 R 等价是微不足道的，并且零次零圈的 Chow 群为零。