Blow-ups of toric varieties

Question

I'm interseted in blow-ups of toric varieties, unfortunately, I don't understand the construction of a blow-up built by a refinement of a fan, if to be more specific, I didn't find any constructions.

For example, let us consider a fan generated by rays (1,0,0), (0,1,0), (0,0,1), (-1,-1,-1), (1,1,1), (-1,0,0). First four rays give us $ \mathbb{C}P ^ 3$. How will our manifold change after adding rays (1,1,1), (-1,0,0) to a fan.

Where can I find something about connection of refinement of a fan with a blow-up of toric variety?

Answer 1

The cone spanned by $(1,0,0), (0,1,0), (0,0,1)$ corresponds to a torus fixed point (which is the origin of $\mathbb{C}^3$). Subdividing this cone by adding the ray $(1,1,1)$ (the sum of the ray generators of this cone) corresponds to blowing up this $\mathbb{C}^3$ chart at the origin. Adding the ray $(-1,0,0)$ similarly blows up at the origin of another chart.

In general a maximum dimensional cone in a fan corresponds to torus fixed point. If this cone is simplicial, subdividing it by adding the sum of the rays spanning a cone we obtain the blow up at this fixed point.

As for a reference I know this is in Fulton’s book Introduction to Toric Varieties. I don’t have my copy of the book handy to give an exact reference. This is also discussed in Chapter 3 of the book by Cox, Little, and Schenck(see Proposition 3.3.15). Afterwards a similar construction for blowing up along larger dimensional torus invariant subvarieties is also given.