Let $\pi:S \rightarrow \mathbb P^2$ be a blow-up at nine points in general position. I am finding an ample divisor $L$ on $S$ of degree two ($L^2=2$). Since $Pic(S) = \mathbb Z h \oplus \mathbb Z e_1 \oplus \mathbb Z e_2 \oplus \cdots \oplus \mathbb Z e_9$, where $h$ is the pullback of $\mathcal O_{\mathbb P^2}(1)$ and $e_1, \cdots, e_9$ are the exceptional divisors, one can try $$L=a h - b_1 e_1 - b_2 e_2 - \cdots - b_9 e_9$$ with $a, b_i \in \mathbb N$ and impose the condition that $L^2 =2$ and $L\cdot E >0$ for any (-1) curve $E \sim a' h - b_1' e_1 - b_2' e_2 - \cdots - b_9' e_9 $.
For example, let $$L = 17h - 8 e_1 - 8e_2 - 5e_3 -5e_4 -5e_5-5e_6-5e_7-5e_8-3e_9.$$ By mere computation with computer, I checked $L \cdot E >0$ for $a' < 2000$. The problem is that there are infinitely many (-1)-curves on $S$. So this naive approach is not so effective.
Here is my question:
Is there an ample divisor of degree two on $S$ ? And if so, how can one find it?
In the case that the answer to the above question is negative, I would like to ask more general question:
Is there an ample divisor of degree two on some relatively minimal rational elliptic surface?
