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I hope that the following example, in the vein of Daniel's comment, can be useful for you.

Let $H \subset \mathbb{P}^3$ be a fixed plane and let us consider the statement:

"A general line $L \subset \mathbb{P}^3$ intersect $H$ in a single point".

One can think of it in the following way: the Grassmannian of lines $\mathbb{G}(1,3)$, via the Plucker embedding, can be identified with a quadric $X \subset \mathbb{P}^5$. The lines contained in $H$ give a $2$-plane $\Pi_H \subset X$. Therefore the word "general" in the statement precisely means

"a line corresponding to a point in the open dense subset $X \setminus \Pi_H \$ of $X$".

Analogously, the lines containing a point $p \in \mathbb{P}^3$ form a $2$-plane $\Pi_p \subset X$. Then in the statement

"A general line $L \subset \mathbb{P}^3$ does not contain the point $p$"

the word "general" precisely means

"a line corresponding to a point in the open dense subset $X \setminus \Pi_p \$ of $X$".

1

I hope the following example, in the vein of Daniel's comment, can be useful for you.

Let $H \subset \mathbb{P}^3$ be a fixed plane and let us consider the statement:

"A general line $L \subset \mathbb{P}^3$ intersect $H$ in a single point".

One can think of it in the following way: the Grassmannian of lines $\mathbb{G}(1,3)$, via the Plucker embedding, can be identified with a quadric $X \subset \mathbb{P}^5$. The lines contained in $H$ give a $2$-plane $\Pi_H \subset X$. Therefore the word "general" in the statement precisely means

"a line corresponding to a point in the open dense subset $X \setminus \Pi_H \$ of $X$".

Analogously, the lines containing a point $p \in \mathbb{P}^3$ form a $2$-plane $\Pi_p \subset X$. Then in the statement

"A general line $L \subset \mathbb{P}^3$ does not contain the point $p$"

the word "general" precisely means

"a line corresponding to a point in the open dense subset $X \setminus \Pi_p \$ of $X$".