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](http://books.google.ca/books?id=CbmaGqCRbhUC&lpg=PP1&pg=PA71#v=onepage&q&f=truepart).


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)



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


