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Derived categories and rationality of conic bundles

Part of: Curves (Co)homology theory Surfaces and higher-dimensional varieties Birational geometry

Published online by Cambridge University Press:  07 October 2013

Marcello Bernardara  and
Michele Bolognesi
Marcello Bernardara
Affiliation:
Univeristät Duisburg–Essen, Fakultät für Mathematik. Universitätstr. 2, 45117 Essen, Germany email mbernard@math.univ-toulouse.fr Institut de Mathématiques de Toulouse (IMT), 118 route de Narbonne, Université Paul Sabatier, F-31062 Toulouse Cedex 9, France email mbernard@math.univ-toulouse.fr
Michele Bolognesi
Affiliation:
IRMAR, Université de Rennes 1, 263 Av. Général Leclerc, Bat. 22, 35042 Rennes Cedex, France email michele.bolognesi@univ-rennes1.fr
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Abstract

We show that a standard conic bundle over a minimal rational surface is rational and its Jacobian splits as the direct sum of Jacobians of curves if and only if its derived category admits a semiorthogonal decomposition by exceptional objects and the derived categories of those curves. Moreover, such a decomposition gives the splitting of the intermediate Jacobian also when the surface is not minimal.

Keywords

conic bundlesrational threefoldsderived categoriessemiorthogonal decompositionintermediate Jacobian

MSC classification

Secondary: 14E08: Rationality questions 14F05: Sheaves, derived categories of sheaves and related constructions 14H40: Jacobians, Prym varieties 14J30: $3$-folds
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 11 , November 2013 , pp. 1789 - 1817
Copyright
© The Author(s) 2013 

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