Derived categories and rationality of conic bundles
arXiv:1010.2417
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.
New version; now also the case of cubic degeneration in P^2 is described in detail. 23 Pages