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 $X$".
