We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
$\text{m}$
-Primary Ideals
Two formulas for the multiplicity of the fiber cone 
$F\left( I \right)\,=\,\oplus _{n=0}^{\infty }{{I}^{n}}/\text{m}{{I}^{n}}$ of an $\text{m}$-primary ideal of a 
$d$-dimensional Cohen–Macaulay local ring 
$\left( R,m \right)$ are derived in terms of the mixed multiplicity 
${{e}_{d-1}}\left( \text{m }|I \right)$, the multiplicity 
$e\left( I \right)$, and superficial elements. As a consequence, the Cohen–Macaulay property of 
$F\left( I \right)$ 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 $\text{m}$-primary ideals with a $d$-generated minimal reduction 
$J$ satisfying 
$\ell \left( {{I}^{2}}/JI \right)\,=\,1$ or 
$l\left( Im \right)Jm) = 1$.
