On the depth of blow-up rings of ideals of minimal mixed multiplicity

We show that if $(R,\text{m})$ is a Cohen-Macaulay local ring and $I$ is an ideal of minimal mixed multiplicity, then $\text{depth}G(I)\ge d-1$ implies that $\text{depth}F(I)\ge d-1$. We use this to show that if $I$ is a contracted ideal in a two dimensional regular local ring then $\depth R[It]-1= \depth G(I) = \depth F(I)$\mathrm{.}Wealsogiveaninfiniteclassofidealswhere$R[It]$ is Cohen-Macaulay but $F(I)$ is not.