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.
In this paper we investigate the divisor C14 inside the moduli space of smooth cubic hypersurfaces in P 5 , whose general element is a smooth cubic containing a smooth quartic rational normal scroll. By showing that all degenerations of quartic scrolls in P 5 contained in a smooth cubic hypersurface are surfaces with one apparent double point, we prove that every cubic hypersurface contained in C14 is rational. Combining our proof with the Hodge theoretic definition of C14, we deduce that on a smooth cubic fourfold every class T ∈ H 2,2 (X, Z) with T 2 = 10 and T · h 2 = 4 is represented by a (possibly reducible) surface of degree four which has one apparent double point. As an application of our results and of the construction of some explicit examples, we also prove that the Pfaffian locus is not open in C14.
We isolate a class of smooth rational cubic fourfolds X containing a plane whose associated quadric surface bundle does not have a rational section. This is equivalent to the nontriviality of the Brauer class β of the even Clifford algebra over the K3 surface S of degree 2 arising from X. Specifically, we show that in the moduli space of cubic fourfolds, the intersection of divisors C 8 ∩ C 14 has five irreducible components. In the component corresponding to the existence of a tangent conic, we prove that the general member is both pfaffian and has β nontrivial. Such cubic fourfolds provide twisted derived equivalences between K3 surfaces of degrees 2 and 14, hence further corroboration of Kuznetsov's derived categorical conjecture on the rationality of cubic fourfolds.
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