# Only five smooth projective toric Fano surfaces

The past few weeks I've been dabbling in some toric geometry, and the following result in toric geometry puzzled me when I first read it:

The only smooth projective toric Fano surfaces are $\mathbb{P}^2,\mathbb{P}^1\times\mathbb{P}^1,\mathrm{Bl}_1\mathbb{P}^2,\mathrm{Bl}_2\mathbb{P}^2,\mathrm{Bl}_3\mathbb{P}^2$.

If you are as ignorant about toric geometry as I am (i.e. you've looked at Cox--Little--Schenck's marvellous book but not much more than that) this statement might be a bit puzzling. Because as you might know: the *del Pezzo surfaces* (a shorthand term for smooth projective Fano varieties of dimension 2) are classified, and these are exactly the surfaces $\mathbb{P}^2,\mathbb{P}^1\times\mathbb{P}^1,\mathrm{Bl}_1\mathbb{P}^2,\ldots,\mathrm{Bl}_9\mathbb{P}^2$, where the points are chosen sufficiently general.

So why did this fact puzzle me? Well: there is a perfectly good notion of blowing up a toric variety. Hence why do we not get $\mathrm{Bl}_4\mathbb{P}^2$ as a toric del Pezzo surface? Surely it is del Pezzo, and via a blow-up we can realise it as a toric surface, right?!

No! Think about this for a while. If you don't know this: a toric blow-up is not an arbitrary blow-up, it is a *blow-up in a torus-invariant point*. This is really the crucial point.

In the case of $\mathbb{P}^2$ there are 3 torus-invariant divisors, and they form a triangle of $\mathbb{P}^1$'s in $\mathbb{P}^2$. If you do a toric blow-up you blow up in one of the intersections. You get a square of $\mathbb{P}^1$'s in $\mathrm{Bl}_1\mathbb{P}^2$, hence there are now 4 intersections to choose from for your toric blow-up, *but only two of them come from $\mathbb{P}^2$*. If you blow-up one of those two original points it was as if you performed a simultaneous blow-up, to get $\mathrm{Bl}_2\mathbb{P}^2$, but if you pick one of the other 2 points, which lie on the exceptional divisor, you are getting something different.

Hence, this is why you cannot get $\mathrm{Bl}_4\mathbb{P}^2$ as a toric del Pezzo surface: it is not toric! It is all completely trivial, but it was a nice thing to realise for me, and what is better than sharing your ignorance with the world?