Let \(R={\textbf{k}}[x_1,\dots ,x_d]\) be a polynomial ring over a field \({\textbf{k}}\) and \(I=(f_1,\dots ,f_{d+1})\) be a height two perfect ideal which is linearly presented. Further we suppose that the ideal I satisfies \((\text {G}_{d-2})\) but neither satisfies \((\text {G}_{d-1})\) nor \((\text {G}_d)\) . The \((\text {G}_{s})\) bounds the minimal number of generators of \(I_{\mathfrak {p}}\) by the height \(\operatorname {ht}{\mathfrak {p}}\) for all primes \({\mathfrak {p}}\) up to height \(s-1\) . In this setting, we provide explicit formulas for the defining ideal of the Rees algebra of I, denoted by \({\mathcal {R}}(I)\) . We further demonstrate that \({\mathcal {R}}(I)\) is not always Cohen-Macaulay ring. While the attempts to find the defining ideal of the Rees algebra ideals satisfying \((\text {G}_{d})\) or \((\text {G}_{d-1})\) conditions are well documented, the \((\text {G}_{d-2})\) case is still unexplored.
