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Let U be the tautological subbundle on the Grassmannian $\mathrm{Gr}(k, n)$.There is a natural morphism $\mathrm{Tot}(U) \to \mathbb{A}^n$. Using it, wegive a semiorthogonal decomposition for the bounded derived category$D^b_{\mathrm{coh}}(\mathrm{Tot}(U))$ into several exceptional objects andseveral copies of $D^b_{\mathrm{coh}}(\mathbb{A}^n)$. We also prove a globalversion of this result: given a vector bundle $E$ with a regular section $s$,consider a subvariety of the relative Grassmannian $\mathrm{Gr}(k, E)$ of thosesubspaces which contain the value of $s$. The derived category of thissubvariety admits a similar decomposition into copies of the base and the zerolocus of $s$. This may be viewed as a generalization of the blow-up formula ofOrlov, which is the case $k = 1$.

Semiorthogonal decompositions on total spaces of tautological bundles