Let $X$ be a smooth projective variety and $\mathcal{E}$ a locally free sheaf of rank $r$ on it. Now we consider the relative Grassmannian $\mathrm{Grass}_X(l,\mathcal{E})$. What is the canonical bundle of the relative Grassmannian?
2026-03-28 15:26:32.1774711592
canonical bundle of relative Grassmannian
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If you know the calculation of the canonical bundle of the usual Grassmannian (nice writeup here), you can adapt it directly to the relative situation. The result is that $K_G=\mathcal O_{\mathbf P}(-r)_{|G}$, where $\mathbf P=\mathbf P_X \left( \bigwedge^l \mathcal E \right)$ is the projective bundle into which the relative Grassmannian is embedded by the Plücker embedding.