Derived categories and rationality of conic bundles.

@inproceedings{Bernardara2013DerivedCA,
  title={Derived categories and rationality of conic bundles.},
  author={Marcello Bernardara and Michele Bolognesi},
  year={2013}
}
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. 

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