Consider the (symplectic) blow up $\operatorname{Bl}_k(\mathbb{CP}^2)$ of $\mathbb{CP}^2$ at $k$ points. I have heard that for $k=1,2,\ldots,8$ the size of the balls been blown up can be choosen in such a way that the symplectic form in $\operatorname{Bl}_k(\mathbb{CP}^2)$, is exact when restricted to the complement of the exceptional divisors.
On contrast with this, I have also heard that no matter the sizes of the balls been blown up for $k>8$, the symplectic form in $\operatorname{Bl}_k(\mathbb{CP}^2)$, when restricted to the complement of the exceptional divisors, will never be exact.
That being said:
1- I would appreciate if somebody could indicate references where both these results are proven.
2- I would also like to understand how the second fact above and the following argument do not contradict each other: Suppose that we do a "almost toric blow up" of $\mathbb{CP}^2$ at 8 different points over the same excepcional divisor (all points over the same edge when regarding the toric picture of $\mathbb{CP}^2$ as a triangle in the plane), with the blown up balls being small as we "need" and all of the same size. Then do a ninth blow up at a point in a different exceptional divisor, also as small as we need. This all can be done in such a form that the hypothesis of Theorem 8.3.1 and Corollary 8.3.2 (in Jonny Evans' notes https://arxiv.org/pdf/2110.08643.pdf), studying the symplectic form in the complement of the exceptional divisors, are met. Which would indicate that even for $k=9$ we can blow up in the "right way" to get an exact symplectic form when restricted to the complement of the exceptional divisors. Contradiction!
Technical note: Theorem 8.3.1 referred to above, assumes a hypothesis that is not exactly met in my 8-1 blow up example, however when I go over the proof I do not see any reason why it shouldn't hold in this case too. But I'm almost sure that my lack of understanding on why I cannot do this improvement to the proof of the theorem is what is leading to the contradiction described above.
