Description of a birational map, Fulton's "Introduction to toric varieties"

Question

This is an exercise of "Introduction to toric varieties" by Fulton, page 71, section 3.4, (the last one in this page.)

The problem is initiated by constructing a complete (toric) variety which is not projective, by taking the fan $\Delta$ whose edges in $\mathbb{Z}^3$ are passing through $v_1=-e_1, v_2=-e_2, v_3=-e_3, v_4=e_1+e_2+e_3, v_5=v_3+v_4, v_6=v_1+v_4$ and $v_7=v_2+v_4$ where $e_i$'s are the standard basis of $\mathbb{Z}^3,$ and with cones through the faces of the triangulated tetrahedron shown, here.

It can be shown that no strictly convex, piece-wise linear, integral function exists over $\Delta,$ which is done in the first exercise. My question is about the next exercise as follows;

1. Describe the birational map from this variety (constructed from the above fan) $X=X(\Delta)$ to $\mathbb{P}^3$ determined by this subdivision (described in the picture of the book) of the pyramid (generated by $v_1,v_2,v_3,v_4.$) In particular, show that the blowing up occures over a plane triangle in $\mathbb{P}^3.$

2. Show that the toric variety obtained by truncating the pyramid and omitting $v_4$ has a singular point of multiplicity $2.$

For 1) intuitively, I can convince myself that there is a $\mathbb{P}^2 \subset \mathbb{P}^3$ (the orbit of the ray $v_4$!) s.t. the birational map blows up the three $T$-invariant curves (associated with orbits of the faces generated by the $v_4, v_3$ and $v_4, v_1$ and $v_4, v_2.$) Each three dimensional cone $\sigma$ is subdivided by two star subdivisions with center rays in two dimensional faces.

So what happens for each $U_{\sigma}=\mathbb{A}^3$? and how can I verify my intuition with explicit computations in each step?

For 2) By omitting the vertex $v_4$ the (simplicial) cone generated by $v_5,v_6,v_7$ is singular of index $2$ (by determinant argument) while the rest of cones are non-singular, so the resulting toric variety has a singular point of multiplicity $2.$

Is that a correct argument?

Updated: For 1) I have this course (An introduction to toric varieties) with Kalle Karu this term. He said that the star subdivisions correspond to the blowups of $U_{\sigma}=\mathbb{A}^3$ in one coordinate axis and in the strict transform of the other, and in different charts the blowups are performed in different order! which I will be very grateful if someone can elaborate it more (with explicit computations, of course)

Answer 1

The multiplicity argument is correct. The multiplicity of a siplicial cone is the order of the class group of the corresponding affine variety.

In general, the explicit calculation question is a bit trickier. If I recall correctly I identify which ideal is being blown up in some cases in arXiv:math/0310336v1.

It might be better to write down the ring elements corresponding to the generators of the dual cone for each cone in the fan of X and work backwards.