Blow-up toric varieties.

Question

I have to take a talk of an hour and I have to talk about blow-up of toric varieties. Can you suggest me some interesting examples that I can present? How can I find a good reference for the theory needed? I'd like to do a talk rich of images and interesting examples. Thank you!

Answer 1

Off the top of my head, here is something you could try. (Everything you need should be in Fulton's book or Cox–Little–Schenck.)

1. Explain how blowing up a subvariety of a toric variety corresponds to subdivision of the fan. Illustrate with simple examples, like blowing up a point in $\mathbf P^2$ and a line in $\mathbf P^3$.

2. Now use this to explain how to resolve the birational map $\sigma: \mathbf P^2 \dashrightarrow \mathbf P^2$ defined by $$\sigma([x_0,x_1,x_2]) = [x_1x_2,x_0x_2,x_0x_1].$$ (Punchline: you subdivide the standard fan in $\mathbf R^2$ in a certain way, then unsubdivide (?) in a different way to get a different, but isomorphic, fan.)

3. Next consider the analogous birational map $\Sigma: \mathbf P^3 \dashrightarrow \mathbf P^3$ defined by $$\Sigma([x_0,x_1,x_2,x_3]) = [x_1x_2x_3,x_0x_2x_3,x_0x_1x_3,x_0x_1x_2].$$ Now there are lots of possibilities: you can blow up lines in different orders, and compare the result. Or you can blow up points to get a toric variety $X$, and study the birational map $\Sigma_X: X \dashrightarrow X$ induced by $\Sigma$. What properties does $\Sigma_X$ have? (Punchline: congratulations, you've invented flops!)

I'm deliberately being a bit sketchy here; I encourage you to work out the details of what I'm talking about. I hope that helps. Have fun!