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 two 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 of X. Together with new results in the theory of quadratic forms, we apply these tools in the case where X -> Y has relative dimension 1, 2, or 3, in which case the fibers are curves of genus 1, Del Pezzo surfaces of degree 4, or Fano threefolds, 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.

43 pages, changes made and some material added and corrected in sections 1, 4, and 5; this is the final version accepted for publication at Journal de Mathématiques Pures et Appliquées

References in corpus (2)

• Gauss composition over an arbitrary base
• Exceptional collections of line bundles on projective homogeneous varieties

Cited by in corpus (15)

• Semiorthogonal decompositions and birational geometry of del Pezzo surfaces over arbitrary fields
• Grothendieck ring of varieties, D- and L-equivalence, and families of quadrics
• Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties
• The Brauer group of the moduli stack of elliptic curves
• Derived Equivalent Calabi-Yau 3-folds from Cubic 4-folds
• Symmetric locally free resolutions and rationality problems
• Retour sur l'arithmétique des intersections de deux quadriques, avec un appendice par A. Kuznestov
• Albert algebras over Z and other rings
• Quadric bundles and hyperbolic equivalence
• On derived categories of arithmetic toric varieties
• A threefold violating a local-to-global principle for rationality
• Residual categories of quadric surface bundles
• Derived Categories of Quintic Del Pezzo Fibrations
• Derived categories of quartic double fivefolds
• The Intermediate Jacobian fibration of a cubic fourfold containing a plane and fibrations in Prym varieties