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# On Fiber Cones of $\m$-Primary Ideals

Published:2007-02-01
Printed: Feb 2007
• A. V. Jayanthan
• Tony J. Puthenpurakal
• J. K. Verma
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## Abstract

Two formulas for the multiplicity of the fiber cone $F(I)=\bigoplus_{n=0}^{\infty} I^n/\m I^n$ of an $\m$-primary ideal of a $d$-dimensional Cohen--Macaulay local ring $(R,\m)$ are derived in terms of the mixed multiplicity $e_{d-1}(\m | I)$, the multiplicity $e(I)$, and superficial elements. As a consequence, the Cohen--Macaulay property of $F(I)$ when $I$ has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of the reduction number of $I$ and lengths of certain ideals. We also characterize the Cohen--Macaulay and Gorenstein properties of fiber cones of $\m$-primary ideals with a $d$-generated minimal reduction $J$ satisfying $\ell(I^2/JI)=1$ or $\ell(I\m/J\m)=1.$
 Keywords: fiber cones, mixed multiplicities, joint reductions, Cohen--Macaulay fiber cones, Gorenstein fiber cones, ideals having minimal and almost minimal mixed multiplicities
 MSC Classifications: 13H10 - Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 13H15 - Multiplicity theory and related topics [See also 14C17] 13A30 - Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13C15 - Dimension theory, depth, related rings (catenary, etc.) 13A02 - Graded rings [See also 16W50]