Segments seem to cause trouble. For non-segments, it seems like there is a bijection between the facets of P+Q, and the union of the facets of P and of Q. (Though if P and Q both have facets with the same outer normal, they form one segment in P+Q.)

Very cool, thank you! The example on p. 71 of Fulton is simple enough that just by staring at it, you can also see that there is no way to go from the coarse fan to the refined fan just by blowups.

Could you say a little bit more? Maybe a reference? Or an example? Or a reason it should be obvious?

@AndrewHubery very cool, thanks!

@AndrewHubery How do you know the representation corresponding to the highest root will be in the AR translation orbit of the projective at the central vertex?

@Kimball I don't think you can actually get "whatever you want" with integer-coefficient polynomials. For example, you can't have f(0)=f(1)=0, f(2)=1.

Beautiful argument!