Phantoms and strongly exceptional collections

Question

I would like to ask a question about phantom categories and strongly exceptional collections.

Let $X$ be a smooth prpjective variety over an algebraically closed field. An admissible subcategory $B \subset D^{b}(X)$ is said to be a phantom category if $K(B) = 0$ but $B \neq 0$ (where $K(B)$ is the K-theory of $B$).

I know (for instance from here : https://arxiv.org/pdf/1209.6183.pdf) that there exists smooth projective varieties having an exceptional collection of vector bundles $E_1, \ldots, E_n$ such that $n = \textrm{rank}(K(X))$, but the collection is not full. This implies in particular that the left orthogonal to the category generated by the $E_i$ is a phantom category.

I was wondering if there exists examples as above, but with the additional assumption that the collection $E_1,\ldots,E_n$ is strongly exceptional? (that is the higher $\mathrm{Ext}$ between the $E_i$'s vanish in both directions).

In all examples I have been reading so far, the exceptional collections of length equal to $\mathrm{rank}(K(X))$ which are not full are never strongly exceptional. Does someone know an example where this could be the case?

Thanks in advance for your answers!

Answer 1

In the case of exceptional collection consisting of line bundles, one can say something about it(at least on surface).

Let $X$ be a smooth projective surface with strong exceptional collection of line bundles of maximal length(i.e length $l=rk(K_0(X))$, then $X$ is a rational surface.

Then in the case of line bundles, your question becomes whether there is an example of a rational surface admitting a strong exceptional collection of line bundle which is not full.

Probably not. L.Hille and M.Perling ("Exceptional sequences of invertible sheaves on rational surfaces", Compositio Math. 147 (2011), 1230-1280) introduced a very natural operations on producing exceptional collection of line bundles on rational surface, called augmentations which is a derived categorical analogue of "blow up". They have the following conjecture:

Let $X$ be a rational surface, then any strong exceptional collection of line bundles is coming from augmentations from $\mathbb{P}^2$ and $\mathbb{F}_n,n\neq 1$.

Since any exceptional collection of line bundles on $\mathbb{P}^2$ and $\mathbb{F}_n$ is full. Thus if the conjecture of Hille-Perling is true, then any strong exceptional collection of line bundles of maximal length on rational surface is full.

The conjecture has been verified for several cases: Toric surface, del Pezzo surface and weak del Pezzo surfaces of degree $K_X^2\geq 3$ and any anticanonical rational surface. Thus, such example in your question can not exist on those varieties.

Unfortunately, the conjecture above is NOT true. There is an example of weak del Pezzo surface of degree $2$ admitting a strong exceptional collection of line bundles of maximal length but not coming from augmentation in any sense. However, by a result of Kuleshov, it still full!

By a computer algorithm checking, the number of counter examples of above conjecture is really small. Thus one may say, in the most cases, strong exceptional collection of line bundles of maximal length is full.

In fact, one is able to show that the so called cyclic strong exceptional collection of line bundles of maximal length is full. I and my collaborators are working to prove that any strong exceptional collection of line bundles on rational surface is full by a completely different method.