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

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

2011-09-30

arxiv.org/abs/1109.6938v1

Mathematics

Algebraic Geometry

33 pages, comments welcome

Scientific paper

Let X -> Y be a fibration whose fibers are complete intersections of r quadrics. We develop some categorical and algebraic tools, such as 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 of X. Together with input from 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 elliptic curves, Del Pezzo surfaces of degree 4, or Fano three-folds, respectively. In the latter two cases, if Y is the projective line over an algebraically closed field of characteristic zero, we relate rationality questions to categorical representability of X.

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