# Show that a generic line is not a leaf of an algebraic foliation of $\mathbb P^2$

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

1. A generic point $$p \in \mathbb P^2$$ lies in the affine part $$\mathbb C^2$$.

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?

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.
• This would work if the foliation had no singularities. But I think algebraic foliations of $\mathbb P^2$ necessarily have singularities. (Incidentally, now I realize that my argument is broken for the same reason.) – pyon Nov 19 '19 at 1:54
• A foliation of $\mathbb P^2$ consists of the following data: (0) an open cover of $\mathbb P^2$, (1) for each open $U$ in the cover, a vector field $X_U$ on $U$, such that (2) for every pair of opens $U, V$ in the cover, the vector fields $X_U$ and $X_V$ are scalar multiples of one another on the overlap $U \cap V$. – pyon Nov 19 '19 at 3:19