Let \(G=PSL(n,\mathbb {C})\) , T be a maximal torus of G and B be a Borel subgroup of G containing T. Let \(S=\{\alpha _{1},\ldots ,\alpha _{n-1}\}\) be the set of simple roots of G relative to (B, T). Let \(W=N_{G}(T)/T\) be the Weyl group of G relative to T. Let \(\{\lambda _{1},\ldots ,\lambda _{n-1}\}\) be the one-parameter subgroups of T dual to \(\{\alpha _{1},\ldots ,\alpha _{n-1}\}\) . Let \(w_{s,r}\in W^{S\setminus \{\alpha _{r}\}}\) be the minimal element such that the Schubert variety \(X(w_{s,r})\) admits semistable points for the \(\lambda _{s}(G_{m})\) -linearized ample line bundle \(\mathcal {L}(n\omega _{r})\) on \(G_{r,n}\) . Let \(L_{S\setminus \{\alpha _{s}\}}\) be the Levi subgroup of the maximal parabolic \(P_{S\setminus \{\alpha _{s}\}}\) corresponding to \(\alpha _{s}\) , and let \(B_{L_{S\setminus \{\alpha _{s}\}}}=B\cap L_{S\setminus \{\alpha _{s}\}}\) . Then for any irreducible component \(\mathcal {X}_{v}\) of the Hessenberg variety with \(v\in {EMPTY}^{S\setminus \{\alpha _{s}\}}W\) with there are exactly two simple roots \(\alpha _{r}\) , \(\alpha _{t}\) that are made negative by v and \(v^{S\setminus \{\alpha _{r}\}}=w_{s,r}\) , \(n\not \mid rs\) , there is an ample line bundle \(\mathcal {L}(\chi )\) on G/B such that \(\lambda _{s} \backslash \hspace{-3.33328pt}\backslash (\mathcal {X}_{v})^{ss}_{\lambda _{s}}(\mathcal {L}(\chi ))\simeq L_{S\setminus \{\alpha _{s}\}}\times ^{B_{L_{S\setminus \{\alpha _{s}\}}}}\mathbb {P}^{1}\) . Further, if there are exactly three simple roots made negative by v, say \(\alpha _{r},\alpha _{t_{1}}\) and \(\alpha _{t_{2}}\) , then we prove that there is an ample line bundle \(\mathcal {L}(\chi )\) on G/B such that \(\lambda _{s} \backslash \hspace{-3.33328pt}\backslash (\mathcal {X}_{v})^{ss}_{\lambda _{s}}(\mathcal {L}(\chi ))\simeq L_{S\setminus \{\alpha _{s}\}}\times ^{B_{L_{S\setminus \{\alpha _{s}\}}}}(\mathbb {P}^{1}\times \mathbb {P}^{1})\) .
