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Oberseminar SoSe 2011 Zusammenfassung des Vortrags von Marcello Bernardara (Essen)

Derived categories of conic bundles

Let XS be a conic bundle over a rational surface. We can naturally associate to it the sheaf B0 on S of the even parts of the Clifford algebra, and the derived category D (S,B0), which embeds full and faithfully in the derived category D (X). In a joint work with M. Bolognesi, we show that X is rational if and only if D (S,B0) admits a semiorthogonal decomposition by exceptional objects and derived categories of smooth projective curves. In my talk I will give a sketch of the proof and an explicit example.