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?


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.

  • $\begingroup$ 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.) $\endgroup$ – pyon Nov 19 '19 at 1:54
  • $\begingroup$ What is your definition of a foliation and a leaf of that foliation, then? (Also, the proof should work in the case of allowing singularities on the leaves: the condition of intersecting a line but not at some finite collection of specified points is still a general condition, and this applies here because the singularities of any 1-dim leaf would be a 0-dim set, or a finite collection of points.) $\endgroup$ – KReiser Nov 19 '19 at 2:19
  • $\begingroup$ 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 $U \cap V$. $\endgroup$ – pyon Nov 19 '19 at 3:19
  • $\begingroup$ And your definition of a leaf of the foliation? $\endgroup$ – KReiser Nov 19 '19 at 3:26
  • $\begingroup$ A curve formed by pasting integral curves of the vector fields. $\endgroup$ – pyon Nov 19 '19 at 3:29

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