The group generated by spherical twists on a Picard-rank-one K3 surface is free, and this proves transitivity and freeness of braid mutations on full exceptional collections for the four Fano threefolds with such a collection of four vector bundles.
Braid group actions on branched coverings and full exceptional sequences
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abstract
We relate full exceptional sequences in Fukaya categories of surfaces or equivalently in derived categories of graded gentle algebras to branched coverings over the disk, building on a previous classification result of the first and third author. This allows us to apply tools from the theory of branched coverings such as Birman--Hilden theory and Hurwitz systems to study the natural braid group action on exceptional sequences. As an application, counterexamples are given to a conjecture of Bondal--Polishchuk on the transitivity of the braid group action on full exceptional sequences in a triangulated category.
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Groups generated by spherical twists on K3 surfaces and full exceptional collections on Fano threefolds
The group generated by spherical twists on a Picard-rank-one K3 surface is free, and this proves transitivity and freeness of braid mutations on full exceptional collections for the four Fano threefolds with such a collection of four vector bundles.