关于纤维锥的深度和Hilbert级数

Let (R,m) be a Cohen-Macaulay local ring of dimension d with infinite residue field, I an m-primary ideal and K an ideal containing I. Let J be a minimal reduction of I such that, for some positive integer k, KIn ∩ J = JKIn-1 for n ≤ k ? 1 and λ( JKKIIkk-1 ) = 1. We show that if depth G(I) ≥ d-2, then such fiber cones have almost maximal depth. We also compute, in this case, the Hilbert series of FK(I) assuming that depth G(I) ≥ d - 1.

On the Depth and Hilbert Series of the Fiber Cone

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 $k$, $KI^ncap J=JKI^{n-1}$ for $nle k-1$ 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 $G(I)ge d-1$.