On the Mixed Multiplicities of Multi-graded Fiber Cones

Abstract

Let $(A,\frak{m})$ denote a Noetherian local ring with maximal ideal $\frak{m}$, $J$ an $\frak m$-primary ideal, $I_1,\ldots, I_s$ ideals of $A$; $M$ a finitely generated $A$-module. This paper will answer when mixed multiplicities of the multi-graded fiber cone $$ F_M(J,I_1,\ldots, I_s)=\bigoplus_{n_1,\ldots,n_s\geqslant 0}\dfrac{I_1^{n_1}\cdots I_s^{n_s}M}{JI_1^{n_1}\cdots I_s^{n_s}M} $$ are positive and characterize them in terms of the length of modules.