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Let $X$ be a smooth complex projective variety and let $D^b(X)$ be the bounded derived category of coherent sheaves on $X$. Then the braid group on $\mathrm{dim}\:K_0(X)$ strands acts by mutations on on the set of full exceptional collections in $D^b(X)$.

This action has been shown to be transitive for collections of vector bundles on Fano threefolds with $b_2=1$ and $b_3=0$ (Polishchuk) or for 3-block collections of fixed structure on smooth del Pezzos (Karpov&Nogin).

Does there exist a smooth projective variety such that the action of braid group on the set of full exceptional collections is intransitive?

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    $\begingroup$ First, even for $\mathbb{P}^1$ you need to add the action of shifts to achieve transitivity of the action. Second, as far as I remember, Polishchuk only proves transitivity for exceptional collections of vector bundles; it doesn't imply transitivity (even with shifts taken into account), since there might be exceptional collections consisting of non-locally free sheaves, or even of complexes. $\endgroup$
    – Sasha
    Commented Sep 5, 2018 at 14:50

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