TITLE: What is the fan of the toric blow-up of $\mathbb{P}^3$ along the union of two intersecting lines?
QUESTION [5 upvotes]: 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!

REPLY [3 votes]: Since you already know how to blow up along either ${\mathbb P}^1$ individually, we can concentrate on what's happening nearby the intersection. Which means we can work affinely.
Then the polyhedron for ${\mathbb A}^3$ is the octant $({\mathbb R}_{\geq 0})^3$, and your two lines correspond to two of the three edges.
To blow them up, take a carpenter's plane to those two edges, and shave them off exactly the same amount. The result will have an edge connecting two vertices, one of degree 3, one of degree 4 (the isolated singularity on the blowup). 
In coordinates, the polyhedron is {$(x,y,z) : x,y,z \geq 0, x+y \geq 1, x+z \geq 1$}.
If you want the blowup of ${\mathbb P}^3$ not ${\mathbb A}^3$, also impose $x+y+z \leq N$ for some large $N$ (three will do).