Multi-graded extended Rees algebras of \(\mathfrak m\)-primary ideals (Q860091)

Let \((R, m)\) be a Cohen-Macaulay local ring with infinite residue field \(k\). \(R\) is said to have minimal multiplicity if the equality \(e(R)= \dim_k m/m^2- \dim(R)+ 1\) holds. For \(m\)-primary ideals \(I_1,\dots, I_g\) of \(R\), the multi-graded extended Rees algebra \({\mathcal B}(I_1,\dots, I_g)\) is the graded ring \(R[I_1 t_1,\dots, I_g t_g, t^{-1}_1,\dots, t^{-1}_g]= \bigoplus_{n_1,\dots, n_g\in\mathbb{Z}}(I, t_1)^{n_1}\dots(I_g t_g)^{n_g}\). Let \(N\) be its homogeneous maximal ideal. The author considers the problem of when the ring \({\mathcal B}(I_1,\dots, I_g)_N\) is a Cohen-Macaulay local ring with minimal multiplicity. First, the author shows that if \({\mathcal B}(I_1,\dots, I_g)_N\) is a Cohen-Macaulay local ring with minimal multiplicity, then we have \(g= 1\) or \(g= 2\). Second, when \(g= 2\), some necessary conditions are given for \({\mathcal B}(I_1, I_2)_N\) to be a Cohen-Macaulay local ring with minimal multiplicity. Finally, an example is given showing that \({\mathcal B}(I_1, I_2)_N\) and \({\mathcal B}(I_1)_N\) are Cohen-Macaulay local rings with minimal multiplicity but \(R\) is not the case.