# Why only some del Pezzo are toric?

Let us define smooth del Pezzo surfaces $$dP_r$$ as the blowup of $$r$$ generic points in $$\mathbb{CP}_2$$. One can show that if we request $$dP_r$$ to be Fano, then $$r=0,...,8$$. In theoretical physics literature is common to define also $$dP_9$$, which is the blowup of $$9$$ points in $$\mathbb{CP}_2$$ and refer to this as well as a del Pezzo surface, while as far as I understand mathematicians would call this a generic rational elliptically fibered surface.

Now, the question is why only $$dP_r$$ for $$r=0,1,2,3$$ are toric varieties? This statement is given for example in the appendix A.3 of http://inspirehep.net/record/1707635

For example, consider the fan of $$\mathbb{CP}_2$$. In order to perform the $$i$$-th blow-up, at the level of the fan, we add a new 1-dimensional cone which is representing the exceptional divisor $$E_i$$, associated to the new $$\mathbb{CP}_1$$. Which is the obstruction from keeping adding cones also for $$r>3$$? Why does the toric description fail?

• You are only allowed to blow up torus-invariant points, and for $\mathbb{P}^2$ these lie on the intersections of the 3 torus-invariant divisors, which form a triangle. You can only blow up these 3 points if you want your surface to stay toric. Mar 28 '19 at 16:57
• There is no obstruction to adding cones for r>3: you get some surface obtained from $\mathbb{P}^2$ by blowing up $r$ points but these $r$ points are not in generic position if $r>3$. Mar 28 '19 at 17:39
• The last phrase of my comment should of course say "if you want your surface to stay toric and del Pezzo", as pointed out by @user25309. Mar 28 '19 at 17:54
• I’ve heard physicists calling rational elliptic surface “half K3”. Apr 2 '19 at 13:08

You have no problem for adding cones from a toric surface $$\mathbb{P}^2$$ or any surface obtained by blowing-up points on it.

However, there are only three toric points on $$\mathbb{P}^2$$, so once you have blown-up these $$3$$ points, the toric points that you see on your surface lie on $$(-1)$$-curves, and thus do not correspond anymore to points of $$\mathbb{P}^2$$. You can still blow-up them, and it corresponds to what you expect on the fan point of view, but you no longer get a del Pezzo surface, as there are $$(-2)$$-curves.

In general, if you blow-up more than $$8$$ points of $$\mathbb{P}^2$$ in (very) general points, you get a non-toric surface with only curve of self-intersection $$\ge -1$$ and such that negative curves should be $$(-1)$$-curves (it is in fact a conjecture, see https://arxiv.org/pdf/math/0512631.pdf ), but this is no longer del Pezzo as the anti-canonical divisor is no longer ample, only nef and not big.

• Small clarification to the last line: if you blow up more than 9 points, the anticanonical is not even nef.
– Pop
Apr 1 '19 at 13:53
• For $9$, it is certainly nef, but for $10$, why should it be not nef? Which curve is positive against the canonical divisor? Apr 4 '19 at 12:42
• Dear J\'er\'emy, a nef divisor on a surface always has nonnegative selfintersection. For 9+n points the anticanonical has selfintersection -n, so it cannot be nef.
– Pop
Apr 4 '19 at 21:18
• To answer your second question: If I counted parameters correctly, given 10 points in the plane there is a curve of degree 9 with multiplicity 3 at 8 of the points and multiplicity 2 at the other 2. Its intersection with the anticanonical is $3 \cdot 9 - 8 \cdot 3 - 2 \cdot 2 =-1$.
– Pop
Apr 5 '19 at 6:35
• Thanks for the answer. Apr 6 '19 at 13:54