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1. CJM 2007 (vol 59 pp. 109)

Jayanthan, A. V.; Puthenpurakal, Tony J.; Verma, J. K.
On Fiber Cones of $\m$-Primary Ideals
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
Categories:13H10, 13H15, 13A30, 13C15, 13A02

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