On the Depth and Hilbert Series of the Fiber Cone

Abstract:

      Let $(R,\frak{m})$ be a Cohen-Macaulay local ring of dimension $d$ with infinite residue field, $I$ an $\frak{m}$-primary ideal and $K$ an ideal containing $I$. Let $J$ be a minimal reduction of $I$ such that, for some positive integer $k$, $KI^n\cap J=JKI^{n-1}$ for $n\le k-1$ and $\lambda(\frac{KI^{k}}{JKI^{k-1}})=1$. We show that if depth $G(I)\ge d-2$, then such fiber cones have almost maximal depth. We also compute, in this case, the Hilbert series of $F_K(I)$ assuming that depth $G(I)\ge d-1$.