  This paper describes a paving by affines for regular nilpotent Hessenberg
varieties in all Lie types, namely a kind of cell decomposition that can be
used to compute homology despite its weak closure conditions. Precup recently
proved a stronger result; we include ours because we use different methods. We
then use this paving to prove that the homology of the Peterson variety injects
into the homology of the full flag variety. The proof uses intersection theory
and expands the class of the Peterson variety in the homology of the flag
variety in terms of the basis of Schubert classes. We explicitly identify some
of the coefficients of Schubert classes in this expansion, which is a problem
of independent interest in Schubert calculus.
