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Let $M$ be the moduli space of semistable vector bundles of fixed determinant $L$ and rank $r$ over a smooth curve $X$. Assume that $gcd(r,deg(L))=1$. Let $\mathcal U$ be the universal bundle over $M\times X$ and let $x\in X$. Denote by $\mathcal L=det(\mathcal U)$ and $\mathcal L_x=\mathcal L|_{M\times \{x\}}$.

Since $Pic(M)=\mathbb Z\mathcal D$ where $\mathcal D$ is the determinant line bundle and since the universal bundle is not unique (we can multiply it by a line bundle over $M$, my question is the following: What is the class of $\mathcal L_x$ in $\mathbb Z/r\mathbb Z$? Any reference for that?

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