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We initiate a study of the Gr\"obner geometry of local defining ideals ofHessenberg varieties by studying the special case of regular nilpotentHessenberg varieties in Lie type A, and focusing on the affine coordinate charton $\mathrm{Flags}(\mathbb{C}^n) \cong GL_n(\mathbb{C})/B$ corresponding to thelongest element $w_0$ of the Weyl group $S_n$ of $GL_n(\mathbb{C})$. Our mainresults are as follows. Let $h$ be an indecomposable Hessenberg function. Weprove that the local defining ideal $I_{w_0,h}$ in the $w_0$-chart of theregular nilpotent Hessenberg variety $\mathrm{Hess}(\mathsf{N},h)$ associatedto $h$ has a Gr\"obner basis with respect to a suitably chosen monomial order.Our Gr\"obner basis consists of a collection $\{f^{w_0}_{k,\ell}\}$ ofgenerators of $I_{w_0,h}$ obtained by Abe, DeDieu, Galetto, and the secondauthor. We also prove that $I_{w_0,h}$ is geometrically vertex decomposable inthe sense of Klein and Rajchgot (building on work of Knutson, Miller, andYong). We give two distinct proofs of the above results. We make thisunconventional choice of exposition because our first proof introduces andutilizes a notion of a triangular complete intersection which is of independentinterest, while our second proof using liaison theory is more likely to begeneralizable to the general $w$-charts for $w \neq w_0$. Finally, using ourGr\"obner analysis of the $f^{w_0}_{k,\ell}$ above and for $p>0$ any prime, weconstruct an explicit Frobenius splitting of the $w_0$-chart of$\mathrm{Flags}(\mathbb{C}^n)$ which simultaneously compatibly splits all thelocal defining ideals of $I_{w_0,h}$, as $h$ ranges over the set ofindecomposable Hessenberg functions. This last result is a local Hessenberganalogue of a classical result known for $\mathrm{Flags}(\mathbb{C}^n)$ and thecollection of Schubert and opposite Schubert varieties in$\mathrm{Flags}(\mathbb{C}^n)$.

Geometric vertex decomposition, Gr\"obner bases, and Frobenius splittings for regular nilpotent Hessenberg varieties