Derived categories of quadric fibrations and intersections of quadrics

By Alexander Kuznetsov. 2005

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Abstract 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 ( S 2 W ∗ ) , B 0 ) , where B 0 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 ( S 2 W ) . 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1d144ec025f707be98b7f1deac10c2c446de2b73

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Stability conditions on Kuznetsov components (2017, methodology): “Instead, we use the rational fibration in conics X 99K P3 and Kuznetsov’s theory of derived categories of quadric fibrations [Kuz08] to reinterpret Ku(X) as a semiorthogonal component in the derived category D(P(3),B0) of modules over the associated sheaf of Clifford algebras on P3, see Section 7.”
Instanton bundles on Fano threefolds (2012, background): “The curve C has genus 2 and it is well known that BY ∼= Db(C ) in this case, see [5] or [14].”
Duality in two-dimensional (2,2) supersymmetric non-Abelian gauge theories (2011, methodology): “5 is related to the following equivalences, found by BondalOrlov [19, 14, 20]: For even N D(QS) ∼= 〈CS,O1, .”
Derived categories of coherent sheaves (2006, background): “n classical algebraic geometry. The theory also leads to some extremely interesting derived equivalences between non-commutative varieties. For more details we refer the reader to the original papers [31, 32, 33, 34]. Acknowledgements. I’m very grateful to Antony Maciocia for teaching me about deriv”