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On Fiber Cones of
$\text{m}$
-Primary Ideals
-
- Journal:
- Canadian Journal of Mathematics / Volume 59 / Issue 1 / 01 February 2007
- Published online by Cambridge University Press:
- 20 November 2018, pp. 109-126
- Print publication:
- 01 February 2007
-
- Article
-
- You have access
- Export citation
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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 
$l\left( Im \right)Jm) = 1$.
