Abstract
This paper investigates the geometry of regular Hessenberg varieties associated with the minimal indecomposable Hessenberg space in the flag variety of a complex reductive group. These varieties form a flat family of irreducible subvarieties of the flag variety, encompassing notable examples such as the Peterson variety and toric varieties linked to Weyl chambers. Our first main result computes the closures of affine cells that pave these varieties explicitly, establishing a correspondence between Hessenberg--Schubert varieties and regular Hessenberg varieties in smaller dimensional flag varieties. We also analyze the singular locus of these varieties, proving that all regular Hessenberg varieties are singular outside of the toric case. Specifically, we extend previous results on the singular locus of the Peterson variety to all Lie types. Additionally, we provide detailed descriptions of Hessenberg--Schubert variety inclusion relations, a combinatorial characterization of smooth Hessenberg--Schubert varieties, and simple formulas for their K-theory and cohomology classes. The paper also includes a classification of all singular permutation flags in each regular Hessenberg variety in type A, linking them to combinatorial patterns, and generalizes these findings using root-theoretic data to all Lie types.