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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)

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1 Answer

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.

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Welcome to MO, Howard! –  David Speyer Nov 9 '11 at 0:26
@Howard M Thompson: Thanks for your response. –  Ehsan M. Kermani Nov 9 '11 at 8:28
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