We study Hessenberg varieties in type A via their local defining equations, called patch ideals. We focus on two main classes of Hessenberg varieties: those associated to a regular nilpotent operator and to those associated to a semisimple operator. In the setting of regular semisimple Hessenberg varieties, which are known to be smooth and irreducible, we determine that their patch ideals are triangular complete intersections, as defined by Da Silva and Harada. For semisimple Hessenberg varieties, we give a partial positive answer to a conjecture of Insko and Precup that a given family of set-theoretic local defining ideals are radical. A regular nilpotent Hessenberg Schubert cell is the intersection of a Schubert cell with a regular nilpotent Hessenberg variety. Following the work of the author with Da Silva, Harada, and Rajchgot, we construct an embedding of the regular nilpotent Hessenberg Schubert cells into the coordinate chart of the regular nilpotent Hessenberg variety corresponding to the longest-word permutation in Bruhat order. This allows us to use work of Da Silva and Harada to conclude that regular nilpotent Hessenberg Schubert cells are also local triangular complete intersections. / Thesis / Master of Science (MSc) / Algebraic varieties provide a generalization of curves in the plane, such as parabolas and ellipses. One such family of these varieties are called Hessenberg varieties, and they are known to have connections to other areas of pure and applied mathematics, including to numerical linear algebra, combinatorics, and geometric representation theory. In this thesis, we view Hessenberg varieties as a collection of subvarieties, called coordinate charts, and study the computational geometry of each coordinate chart. Although this is a local approach, we recover global geometric data on Hessenberg varieties. We also provide a partial positive answer to an open question in the area.