Look at the lattice $N= \mathbb{Z}^3/\mathbb{Z}(1,1,2)$, let $u_0,u_1,u_2$ be the images of the standard basis elements of $\mathbb{Z}^3$ and consider the cone $\sigma = \text{Cone}(u_0,u_1)$. Then it is easy to see that this corresponds to the affine variety $V(x_0x_2-x_1^2)$, which is singular at zero. Now one can blow up this affine variety at the coordinate functions, and get a smooth variety (the blow up just inserts a $\mathbb{P}^1$ at the zero point). I have read that the corresponding projection of that blow up map is the same as the the map induces by the identity on $N$ between the toric varieties obtained by the fan consisting of $\sigma$ and the one consisting of the refinement $\sigma_1 = \text{Cone}(u_0,-u_2)$ and $\sigma_2 = \text{Cone}(u_1,-u_2)$. But I am unable to see this, in particular, I do not know how I could calculate the fiber of that second map over zero.
Can someone explain that to me or give me a reference?