Is there a good way to find the fan and polytope of the blow-up of $\mathbb{P}^3$ along the union of two invariant intersecting lines?


Everything I find in the literature is for blow-ups along smooth invariant centers.


Thanks!



What is the fan of the toric blow-up of $\mathbb{P}^3$ along the union of two intersecting lines?
The question is local near the intersection of the two lines, so the more basic question is to work this out for $\mathbb{A}^3$. 


So we want to blow up $k[x,y,z]$ at $\langle z, xy \rangle$. There are two maximal charts:


$$\mathrm{Spec} \ k[x,y,z, (xy)/z] \ \mbox{and} \ \mathrm{Spec} \ k[x,y,z,z/(xy)].$$


Note that the first chart is isomorphic to $k[a,b,c,d]/(ad-bc)$, so we expect it to correspond to a cone with four rays in the fan, while the second chart is isomorphic to $k[u,v,w]$, so we expect it to give an ordinary simplicial cone.


Let's work out exactly how these fit together. Our first semigroup is the integer points in the cone $\mathrm{Span}\_{\mathbb{R}\_{+}} ((1,0,0), (0,1,0), (0,0,1), (1,1,-1))$. The dual cone is $C\_1 := \{ (u,v,w) : u,v,w \geq 0, \ u+v \geq w \}$. So we expect that cone to appear in the fan. Similarly, the second chart corresponds to the cone $C\_2 := \{ (u,v,w) : u,v \geq 0, \ u+v \leq w \}$. We have omitted the inequality $w \geq 0$ because it is redundant.


So $\mathbb{A}^3$ blown up at two crossing lines in the toric variety for the fan which consists of the two cones $C\_1$ and $C\_2$ above (and their faces).


In order to do $\mathbb{P}^3$, you'll want to insert the picture I just described in place of the cone which corresponds to the torus invariant $\mathbb{A}^3$ around the intersection. You'll also have to modify the two other torus charts which contain one of the lines (each). Since that's a blow up at a smooth center, I'll leave it to you.


