An upper bound on the second fiber coefficient of the fiber cones (Q2907934)

Let \((R, \mathfrak m)\) be a Cohen-Macaulay local ring of dimension \(d>0\), having infinite residue field. Let \(I\) be an \(\mathfrak m\)-primary ideal of \(R\) and \(K\) an ideal containing \(I\). The fiber cone of \(I\) with respect to \(K\) is the standard graded algebra \(F_K(I)=\bigoplus_{n\geq 0} I^n/KI^n\). For \(K=I\), \(F_K(I)=G(I)\), the associated graded ring of \(I\). Under the assumptions that \(\text{depth }G(I)\geq d-1\) and \(r_L(I\,|\,K)<\infty\) (where \(r_L(I\,|\,K)\) is a technical notion defined in Definition 2.2 of the paper), the main result of the paper (Theorem 3.2) gives an upper bound on the second fiber coefficient \(f_2(I,K)\) of \(F_K(I)\), with equality holding if and only if \(\text{depth }F_K(I)\geq d-2\).