A remark on toric foliations (Q6536630)

A toric foliation \(\mathcal{F}\) on a toric variety \(X\) is a nonzero saturated subsheaf \(\mathcal{F}\subseteq \mathcal{T}_X\) that is torus equivariant and closed under the Lie bracket, where \(\mathcal{T}_X\) is the tangent sheaf of \(X\). In the paper under review, the author proves that, for a projective \(\mathbb{Q}\)-factorial toric variety, the length of any extremal ray with respect to a toric foliation \(\mathcal{F}\) does not exceed \(r(\mathcal{F})+1\) where \(r(\mathcal{F})\) is the rank of \(\mathcal{F}\); moreover, if \(\mathcal{F}\) has an extremal ray whose length is greater than \(r(\mathcal{F})\), then the associated extremal contraction is a projective space bundle and \(\mathcal{F}\) is the relative tangent sheaf of the extremal contraction and thus locally free.