An upper bound on the second fiber coefficient of the fibercones

Abstract

Let \((R, \mathfrak{m})\) be a Cohen-Macaulay local ring of dimension \(d > 0\), \(I\) an \(\mathfrak{m}\)-primary ideal of \(R\) and \(K\) an ideal containing \(I\). When \(\operatorname{depth} G(I) \geq d - 1\) and \(r(I|K) < \infty\), we present an upper bound on the second fiber coefficient \(f_2(I, K)\) of the fiber cones \(F_K(I)\), and also provide a characterization, in terms of \(f_2(I, K)\), of the condition \(\operatorname{depth} F_K(I) \geq d - 2\).