Rational pullbacks of toric foliations

The paper proves that R_q(n,X,ē) ⊂ L_q(n,ē) always, and that R_q fills an irreducible component of F_q(P^n, ∑e_i) if and only if X is a weighted projective space or a fake weighted projective space (Theorem 4.10). This uses the explicit logarithmic parameterization ρ(q,ē) of L_q and a tangent-space comparison showing R_q is a proper subvariety of L_q unless m=q+1, in which case X is weighted/fake weighted and equality holds (Definition 4.1, Eq. (4.1), Theorem 4.5, Proposition 4.2, Theorem 4.10). The candidate solution reaches the same if-and-only-if conclusion and the inclusion R_q ⊂ L_q via Cox coordinates and logarithmic forms; it then argues by dimension/counting on the residual coefficients. A minor imprecision is that the solution says the allowed λ vary in Gr(q,V_X); in fact V_X has dimension q, so Λ^q V_X is 1-dimensional and the residual coefficients are fixed (up to scale) by X, consistent with the paper’s “constant b_I” viewpoint. This does not affect the conclusion. Overall, both are correct, with the paper relying on a differential (tangent-space) argument and the model on a (compatible) parameter-counting argument (Definition 4.1 and Remark 4.4; Theorem 4.5; Proposition 4.2; Theorem 4.10).