  We study geometric and topological properties of Hessenberg varieties of
codimension one in the type A flag variety. Our main results: (1) give a
formula for the Poincar\'e polynomial, (2) characterize when these varieties
are irreducible, and (3) show that all are reduced schemes. We prove that the
singular locus of any nilpotent codimension one Hessenberg variety is also a
Hessenberg variety. A key tool in our analysis is a new result applying to all
(type A) Hessenberg varieties without any restriction on codimension, which
states that their Poincar\'e polynomials can be computed by counting the points
in the corresponding variety defined over a finite field. The results below
were originally motivated by work of the authors in [arXiv:2107.07929] studying
the precise relationship between Hessenberg and Schubert varieties, and we
obtain a corollary extending the results from that paper to all codimension one
(type A) Schubert varieties.
