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Let $U$ and $W$ be vector spaces over $\mathbb C$ of dimensions $u < w$. There is a geometric quotient $$p: \mathbb P\mathrm{Hom}(U, W)_\text{inj} \xrightarrow{\mathrm{PGL(U)}} \mathrm{Gr}(u, W),$$ where the projective linear group of $U$ acts freely on injective homomorphisms $U \to W$, and a quotient is the corresponding Grassmannian.

The Picard group of left side is $\mathrm{Pic}(\mathbb P^n_\text{inj})=\mathrm{Pic}(\mathbb P^n)=\mathbb Z \mathcal O(-1)$, because the codimension of non-injective maps is more than one. The Picard group of right side is $\mathrm{Pic}(\mathrm{Gr})=\mathbb Z(\operatorname{det} \mathcal U)$, generated by the top exterior power of the universal subbundle $\mathcal U$. I would like to know whether its inverse image is $p^* \operatorname{det} \mathcal U=\mathcal O (-u)$.


The idea comes from an article Knop, Kraft, Vust "The Picard group of a $G$-variety": the mapping $p: X \xrightarrow{G} \mathrm{Gr}$ from above gives an isomorphism $\mathrm{Pic}(\mathrm{Gr}) \simeq \mathrm{Pic}_G(X)$ (prop. 4.2) and a left exact sequence (lemma 2.2) $$0 \to \mathrm{Pic}_G(X) \to \mathrm{Pic}(X) \to \mathrm{Pic}(G),$$ where $\mathrm{Pic}_G(X)$ denotes $G$-equivariant linear bundles on $X$.

The Picard group of projective linear group is $\mathrm{Pic}(\mathrm{PGL}(U))=\mathbb Z_u$, thus the sequence $$0 \to \mathbb Z(\mathrm{det}\,\mathcal U) \xrightarrow{p^*} \mathbb Z\mathcal O(-1) \to \mathbb Z_u$$ is exact if and only if $p^* \operatorname{det} \mathcal U=\mathcal O (\pm u)$.

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On $\mathbb P\mathrm{Hom}(U,W)_{inj}$ there is a tautological embedding $$ \mathcal O(-1) \to \mathrm{Hom}(U,W)\otimes \mathcal O = U^\vee \otimes W \otimes \mathcal O. $$ It induces a map of vector bundles $U \otimes \mathcal O(-1) \to W \otimes \mathcal O$ which is injective at every point. By the universal property of the Grassmannian it induces a map $\mathbb P\mathrm{Hom}(U,W)_{inj} \to \mathrm{Gr}(u,W)$ such that the pullback of the tautological bundle is $U \otimes \mathcal O(-1)$. Clearly, this map is equal to the quotient map in the question. Thus the pullback of the tautological bundle is $U \otimes \mathcal O(-1)$, and the pullback of its determinant is $$ \det(U \otimes \mathcal O(-1)) \cong \det U \otimes \mathcal O(-u). $$

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