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.