Derived categories of quadric fibrations and intersections of quadrics

We construct a semiorthogonal decomposition of the derived category of coherent sheaves on a quadric fibration consisting of several copies of the derived category of the base of the fibration and the derived category of coherent sheaves of modules over the sheaf of even parts of the Clifford algebras on the base corresponding to this quadric fibration generalizing the Kapranov's description of the derived category of a single quadric. As an application we verify that the noncommutative algebraic variety (P(S2W∗),B0), where B0 is the universal sheaf of even parts of Clifford algebras, is Homologically Projectively Dual to the projective space P(W) in the double Veronese embedding P(W)→P(S2W). Using the properties of the Homological Projective Duality we obtain a description of the derived category of coherent sheaves on a complete intersection of any number of quadrics.