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Hessenberg varieties are subvarieties of the flag variety Flags(C^n), the study of which have rich interactions with symplectic geometry, representation theory, and equivariant topology, among other research areas, with particular recent attention arising from its connection to the famous Stanley-Stembridge conjecture in combinatorics. The special case of regular nilpotent Hessenberg varieties has been much studied, and in this talk I describe some work in progress analyzing the local defining ideals of these varieties. In particular, using some techniques relating liaison theory, geometric vertex decomposition, and the theory of Grobner bases (following the work of Klein and Rajchgot), we are able to show that, for the coordinate patch corresponding to the longest word w_0, the local defining ideal for any indecomposable Hessenberg variety is geometrically vertex decomposable, and we find an explicit Grobner basis for a certain monomial order. This is a report on joint work in progress with Sergio Da Silva.