Vector bundles, semisimple algebras and birational properties of rational surfaces

Abstract. In this talk, I will explain how the notion of semiorthogonal decomposition of the derived category allows us to study birational properties of geometrically rational varieties. In the case where \rho=1, this relies on the study of some vector bundles and their endomorphism algebras. In particular, the surface is k-rational if and only if such algebras are étale. As an application, we can show that a non-k-rational minimal del Pezzo surface of degree 6 is birationally semirigid. All these results are obtained in a work in progress with A.Auel.

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