  We give a characteristic-free proof that general codimension-1 Schubert
varieties meet transversally in a Grassmannian and in some related varieties.
Thus the corresponding intersection numbers computed in the Chow (and quantum
Chow) rings of these varieties are enumerative in all characteristics. We show
that known transversality results do not apply to these enumerative problems,
emphasizing the need for additional theoretical work on transversality. The
method of proof also strengthens some results in real enumerative geometry.
