algebraic geometry – Global description of holomorphic foliations of $mathbb P^n$

The nice thing about working with a projective variety $V subset mathbb P^n$ is that, to study it, you do not need to dance the annoying dance of taking affine charts and applying transition functions all the time. Instead, you work with the system of equations that defines its affine cone $C(V) subset mathbb A^{n+1}$, and remember that you are actually interested in the orbits of the rescaling $mathbb C^star$-action (after removing the origin).

However, I am working with 1-dimensional foliations of $mathbb P^n$, rather than subvarieties of $mathbb P^n$. From what I have seen, such foliations are described in terms of a polynomial vector field on the affine part $mathbb A^n$:

$$mathscr F : P_1 , frac partial {partial x_1} + dots + P_n , frac partial {partial x_n}$$

Suppose I want to get a description of $mathscr F$ on another affine chart of $mathbb P^n$, say,

$$y_1 = frac 1 {x_1}, qquad y_2 = frac {x_2} {x_1}, qquad dots qquad y_n = frac {x_n} {x_1}$$

Then I need to compute the pushforward of the change of coordinates and then apply it to the vector field. It is not a terribly difficult operation to perform, but it is still annoying to have to do it all the time. What I wish I could do is define a single polynomial vector field $X$ on $mathbb A^{n+1}$ such that the differential of the projection $pi : mathbb A^{n+1} setminus { 0 } to mathbb P^n$ sends $X(p)$ to a generator or $mathscr F_{pi(p)}$. In particular, $X(p)$ should be tangent to the $mathbb C^star$-orbit if and only if $pi(p)$ is a singular point of $mathscr F$.

  1. Is this always possible?

  2. If so, how?

  3. If not, why?