  Regular semisimple Hessenberg varieties are a family of subvarieties of the
flag variety that arise in number theory, numerical analysis, representation
theory, algebraic geometry, and combinatorics. We give a "Giambelli formula"
expressing the classes of regular semisimple Hessenberg varieties in terms of
Chern classes. In fact, we show that the cohomology class of each regular
semisimple Hessenberg variety is the specialization of a certain double
Schubert polynomial, giving a natural geometric interpretation to such
specializations. We also decompose such classes in terms of the Schubert basis
for the cohomology ring of the flag variety. The coefficients obtained are
nonnegative, and we give closed combinatorial formulas for the coefficients in
many cases. We introduce a closely related family of schemes called regular
nilpotent Hessenberg schemes, and use our results to determine when such
schemes are reduced.
