Let $X$ be a rational surface. Let $\mathbb{E}:=(E_1,\ldots,E_n)$ be strong exceptional collection of line bundles of maximal length $l=rk Pic(X)+2$ in $D^b(coh(X))$, Is there any example that such collection $\mathbb{E}$ is not full, which means that the semi-orthogonal complement $\mathcal{A}$ such that $<\mathbb{E},\mathcal{A}>=D^b(coh(X))$ and $\mathcal{A}\neq 0$

Further, I wonder is it possible that $D^b(coh(X))$ has phantom category $\mathcal{A}$ when $X$ is rational surface ? It looks like the known examples of Phantom category are surface of general type with $p_g=q=0$.

Another question: I wonder whether is there any method to test whether an exceptional collection is full. There is a paper by A.Kusnetsov https://arxiv.org/pdf/1211.4693.pdf

He gives a necessary condition and a sufficient condition. Unfortunately, our example satisfies his necessary condition and it looks not very easy to check his sufficient condition.

Thanks!