Given a smooth projective variety $X$ over a number field $k$ and $P\in X(k)$, the first author conjectured that in a precise sense, any sequence that approximates $P$ sufficiently well must lie on a rational curve. We prove this conjecture for smooth split toric surfaces conditional on Vojta's conjecture. More generally, we show that if $X$ is a $\mathbb{Q}$-factorial terminal split toric variety of arbitrary dimension, then $P$ is better approximated by points on a rational curve than by any Zariski dense sequence.

中文翻译：

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在复曲面上逼近有理点

给定在数域 $k$ 和 $P\in X(k)$ 上的平滑投影变体 $X$，第一作者推测，在精确意义上，任何足够好地逼近 $P$ 的序列都必须基于有理数曲线。我们在以 Vojta 猜想为条件的光滑分裂复曲面上证明了这个猜想。更一般地，我们表明，如果 $X$ 是任意维度的 $\mathbb{Q}$-阶乘终分裂复曲面变体，那么 $P$ 用有理曲线上的点比用任何 Zariski 稠密序列更好地近似。