Fibrations in complete intersections of quadrics, Clifford algebras, derived categories, and rationality problems

Abstract

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 Db(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=P1 over an algebraically closed field of characteristic zero, we relate rationality questions to categorical representability of X.