We prove a categorical version of the Torelli theorem for cubic threefolds. More precisely, we show that the non-trivial part of a semi-orthogonal decomposition of the derived category of a cubic threefold characterizes its isomorphism class.
Let X → Y be a fibration whose fibers are complete intersections of r quadrics. We develop new categorical and algebraic tools-a theory of relative homological projective duality and the Morita invariance of the even Clifford algebra under quadric reduction by hyperbolic splitting-to study semiorthogonal decompositions of the bounded derived category D b (X). Together with results in the theory of quadratic forms, we apply these tools in the case where r = 2 and X → Y has relative dimension 1, 2, or 3, in which case the fibers are curves of genus one, Del Pezzo surfaces of degree 4, or Fano threefolds, respectively. In the latter two cases, if Y = P 1 over an algebraically closed field of characteristic zero, we relate rationality questions to categorical representability of X.
Abstract. A semiorthogonal decomposition for the bounded derived category (the category of perfect complexes in a non smooth case) of coherent sheaves on a Brauer Severi scheme is given. It relies on bounded derived categories (categories of perfect complexes in a non smooth case) of suitably twisted coherent sheaves on the base.
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