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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!

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  1. One paper which does (edit: not, I missed the word rational) address your first question (by giving an example) is

    Galkin, Sergey; Katzarkov, Ludmil; Mellit, Anton; Shinder, Evgeny, Derived categories of Keum's fake projective planes, Adv. Math. 278, 238-253 (2015). ZBL1327.14081.

    There are other examples, https://arxiv.org/abs/1410.3098v1 lists some in the introduction.

  2. Your second question is an open question I think, except that maybe we know that this is not possible in easy cases (such as $\mathbb{P}^2$, for which transitivity of the braid group action is known).

  3. Your third question is the holy grail in the construction of exceptional collections, and no such thing is known as far as I know.

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  • $\begingroup$ Thanks very much! Your comment for 2 is very helpful, in our examples, we know that the mutations(braid group actions) on set of maximal length exceptional collections is transitive which solved my question in my example. $\endgroup$
    – user41650
    Commented Apr 3, 2017 at 7:07
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    $\begingroup$ An example of a rational surface with a phantom is the recent arxiv.org/abs/2304.01269 $\endgroup$
    – Franco
    Commented Apr 6, 2023 at 11:01

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