Consider a foliation of $\mathbb P^2$, whose leaves in the affine part $z = 1$ are the integral curves of a polynomial vector field
$$X = P \frac \partial {\partial x} + Q \frac \partial {\partial y}$$
How do I prove that a generic line $L \subset \mathbb P^2$ is not a leaf of this foliation?
Suppose I were allowed to fix a generic point $p \in \mathbb P^2$ and only then pick a generic line $L \subset \mathbb P^2$ passing through $p$. Then it is intuitively clear that
A generic point $p \in \mathbb P^2$ lies in the affine part $\mathbb C^2$.
A generic tangent vector $v \in T_p \mathbb C^2$, which plays the role of the tangent direction of $L$, is not parallel to $X(p)$.
But am I allowed to fix a generic point and only then pick a generic line passing through it in the first place? If so, why?
Something about the language of "fixing a generic point" seems unpleasant to me. How about the following solution instead:
If no line in $\Bbb A^2$ is a leaf of the foliation, then a general line in $\Bbb P^2$ is not a leaf, as a general line passes through $\Bbb A^2$. If a line $\ell$ in $\Bbb A^2$ is a leaf of the foliation, then any line intersecting $\ell$ in $\Bbb A^2$ may not be a leaf of the foliation: leaves of the foliation are required to be smooth and non-intersecting. On the other hand, the condition of "intersects $\ell$ in $\Bbb A^2$" is a generic condition on the set of lines in $\Bbb P^2$. So a general line is not a leaf of this foliation.