  Although regular semisimple Hessenberg varieties are smooth and irreducible,
semisimple Hessenberg varieties are not necessarily smooth in general. In this
paper we determine the irreducible components of semisimple Hessenberg
varieties corresponding to the standard Hessenberg space. We prove that these
irreducible components are smooth and give an explicit description of their
intersections, which constitute the singular locus. We conclude with an example
of a semisimple Hessenberg variety corresponding to another Hessenberg space
which is singular and irreducible, showing that results of this nature do not
hold for all semisimple Hessenberg varieties.
