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?

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    $\begingroup$ 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. $\endgroup$
    – pbelmans
    Mar 28 '19 at 16:57
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    $\begingroup$ 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 $r>3$. $\endgroup$
    – user25309
    Mar 28 '19 at 17:39
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    $\begingroup$ 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. $\endgroup$
    – pbelmans
    Mar 28 '19 at 17:54
  • $\begingroup$ I’ve heard physicists calling rational elliptic surface “half K3”. $\endgroup$ 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.

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    $\begingroup$ Small clarification to the last line: if you blow up more than 9 points, the anticanonical is not even nef. $\endgroup$
    – Pop
    Apr 1 '19 at 13:53
  • $\begingroup$ For $9$, it is certainly nef, but for $10$, why should it be not nef? Which curve is positive against the canonical divisor? $\endgroup$ Apr 4 '19 at 12:42
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    $\begingroup$ 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. $\endgroup$
    – Pop
    Apr 4 '19 at 21:18
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    $\begingroup$ 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 $3 \cdot 9 - 8 \cdot 3 - 2 \cdot 2 =-1$. $\endgroup$
    – Pop
    Apr 5 '19 at 6:35
  • $\begingroup$ Thanks for the answer. $\endgroup$ Apr 6 '19 at 13:54

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