While working on a research problem (algebraic cycles), I bumped into a question that I want to prove, though I couldn't yet prove. After several days of attempts, I realized that if the following statement on projective geometry holds, then my original question is most likely answered affirmatively.

Statement:Let $k$ be an infinite perfect field (or even suppose algebraically closed, if you wish). Let $X$ be a smooth projective variety over $k$ of dimension $d$, and let $x_0, x_1, x_2, \cdots$ be an infinite sequence of distinct closed points of $X$. Then, there exist an embedding $X\hookrightarrow \mathbb{P}^N$ and a sequence of hyperplanes $H_1, H_2, \cdots, H_d$, each containing $x_0$, such that $X \cap H_1 \cap \cdots \cap H_d$ is a finite set contained in the given infinite countable set $\{ x_0, x_1, x_2, \cdots \}$.

I think that the statement is true when $\dim X = 1$, simply by taking sufficiently many points $x_0, \cdots, x_{2g+3}$, say, to form a very ample divisor $D = \sum_{i=0} ^{2g+3} x_i$. But, I do not know what to do if $\dim X >1$. Can someone help me in figuring out if this statement is true? I hope I could get some good ideas form mathoverflow. Thank you.