We initiate a study of the Gröbner geometry of local defining ideals of Hessenberg varieties by studying the special case of regular nilpotent Hessenberg varieties in Lie type A, and focusing on the affine coordinate chart on Flags(Cn)≅GLn(C)/B$${\textrm{Flags}}(\mathbb {C}^n) \cong GL_n(\mathbb {C})/B$$ corresponding to the longest element w0$$w_0$$ of the Weyl group Sn$$S_n$$ of GLn(C)$$GL_n(\mathbb {C})$$. Our main results are as follows. Let h be an indecomposable Hessenberg function. We prove that the local defining ideal Iw0,h$$I_{w_0,h}$$ in the w0$$w_0$$-chart of the regular nilpotent Hessenberg variety Hess(N,h)$${\textrm{Hess}}(\textsf{N},h)$$ associated to h has a Gröbner basis with respect to a suitably chosen monomial order. Our Gröbner basis consists of a collection {fk,ℓw0}$$\{f^{w_0}_{k,\ell }\}$$ of generators of Iw0,h$$I_{w_0,h}$$ obtained by Abe, DeDieu, Galetto, and the second author. We also prove that Iw0,h$$I_{w_0,h}$$ is geometrically vertex decomposable in the sense of Klein and Rajchgot (building on work of Knutson, Miller, and Yong). We give two distinct proofs of the above results. We make this unconventional choice of exposition because our first proof introduces and utilizes a notion of a triangular complete intersection which is of independent interest, while our second proof using liaison theory is more likely to be generalizable to the general w-charts for w≠w0$$w \ne w_0$$. Finally, using our Gröbner analysis of the fk,ℓw0$$f^{w_0}_{k,\ell }$$ above and for p>0$$p>0$$ any prime, we construct an explicit Frobenius splitting of the w0$$w_0$$-chart of Flags(Cn)$${\textrm{Flags}}(\mathbb {C}^n)$$ which simultaneously compatibly splits all the local defining ideals of Iw0,h$$I_{w_0,h}$$, as h ranges over the set of indecomposable Hessenberg functions. This last result is a local Hessenberg analogue of a classical result known for Flags(Cn)$${\textrm{Flags}}(\mathbb {C}^n)$$ and the collection of Schubert and opposite Schubert varieties in Flags(Cn)$${\textrm{Flags}}(\mathbb {C}^n)$$.