Bondal-Polishchuk Conjecture for a Smooth Projective Variety
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Statement
Bondal and Polishchuk conjectured in 1993 that the braid group acts transitively on the set of full exceptional collections in a triangulated category. Chang, Haiden and Schroll disproved it for partially wrapped Fukaya categories, but no counterexample of the form for a smooth projective variety was known. A particular weak Fano threefold provides one.
Context
first counterexample of the form D^b(X) for X smooth projective
A 1993 conjecture about exceptional collections that is standard background in derived categories of coherent sheaves and had resisted a smooth projective counterexample.
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The AI disclosure is carefully bounded and worth quoting in full: the model most significantly found numerous mistakes in earlier attempted constructions, leading to the successful modifications, and it located the reference containing the threefold used. The author states that it did not produce the actual counterexample and was not used to generate any of the text.
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