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Regular version of the site

Working paper

Semiorthogonal decompositions on total spaces of tautological bundles

Pirozhkov D. V.
Let U be the tautological subbundle on the Grassmannian Gr(k,n). There is a natural morphism Tot(U)→𝔸^n. Using it, we give a semiorthogonal decomposition for the bounded derived category D^b_coh(Tot(U)) into several exceptional objects and several copies of D^b_coh(𝔸n). We also prove a global version of this result: given a vector bundle E with a regular section s, consider a subvariety of the relative Grassmannian Gr(k,E) of those subspaces which contain the value of s. The derived category of this subvariety admits a similar decomposition into copies of the base and the zero locus of s. This may be viewed as a generalization of the blow-up formula of Orlov, which is the case k=1.