Toric surfaces among rational surfaces I'm learning about rational surfaces. Precisely, I've just read that all minimal rational projective, smooth surfaces are $\mathbb{P}^2$ and the Hirzebruch surfaces $\mathbb{F}_n$, for $n=0,2,3,\ldots$ Here minimal (at least in the algebraic category) means that inside the surface there are no $\mathbb{P}^1$'s with self-intersection equal to $-1$. It is known that every surface dominates (in the sense of birational geometry) a minimal surface and that such minimal model can be obtained contracting these $(-1)$-curves. On the other hand, starting from a given surface, $(-1)$-curves can be produced using the process of blowing-up. It seems to me that blowing-up and contraction are mutual inverse operations.
It turns out that all the above surfaces ($\mathbb{P}^2$ and the $\mathbb{F}_n$) are toric surfaces and I've read that in order to obtain the toric surface $\widetilde{X}$ which is the blow-up of a toric surface $X$ you must include in the fan (a combinatorial object) some rays.
My question is: does it show that every rational (smooth, projective) surface is toric?
If not, it comes the second question: how to distinguish the toric surfaces among the rational ones?
Thanks.

The simples example of a non-toric surface is the blowup of three collinear points on a plane. To see that it is not toric, note that it has a unique $(-2)$-curve (the strict transform of the line through the blowup centers), which intersects exactly three $(-1)$-curves. So, if there is a torus action on this surface, it should preserve this $(-2)$-curve, and fix the three points on it. Hence, the whole $(-2)$-curve should consist of torus fixed points. But a toric surface has only isolated fixed points.
