First Chern class of the universal bundle 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 $\mathcal U_x=\mathcal U|_{M\times\{x\}}$. I read on a paper without any reference or proof that the first chern class $c_1(\mathcal U_x)\in H^2(M,\mathbb Z)=\mathbb Z$ is coprime to $r$. 

How to show that? any reference?  

 A: It was shown in [Remark 2.9] of the paper
The moduli spaces of vector bundles over an algebraic curve, by S Ramannan
https://eudml.org/doc/162339
A: Let $\pi:X\times M \to M$ be the projection.
Theorem 9.11 in Atiyah-Bott's "The Yang-Mills Equations over Riemann Surfaces" states that $c_1(\mathcal{U}_x)$ and $c_1(R\pi_*\mathcal{U})$ generate the second cohomology group (knowing that $M$ is simply connected). This is true an arbitrary choice of $\mathcal{U}$. By Grothendieck-Riemann-Roch, we know that
$$c_1(R\pi_*\mathcal{U}) = (1-g)c_1(\mathcal{U}_x) + \pi_*\operatorname{ch}_2(\mathcal{U}) .$$
It follows that an alternative set of generators for the integral second cohomology is given by $c_1(\mathcal{U}_x)$ and $\pi_*\operatorname{ch}_2(\mathcal{U})$.
Note that tensoring $\mathcal{U}$ by $\pi^*H$ for a line bundle $H$ on $M$ changes $c_1(\mathcal{U})$ by $rc_1(H)$, and that
$$\pi_*\operatorname{ch}_2(\mathcal{U}\otimes \pi^*H) =\pi_*\operatorname{ch}_2(\mathcal{U})+d c_1(H)$$
Now choose integers $a,b$ satisfying $ar - b d = 1$, and pick $H$ satisfying $c_1(H) = \pi_*\operatorname{ch}_2(\mathcal{U})$. Let $\mathcal{U'}:=\mathcal{U}\otimes \pi^*H$.
Then $\pi_*\operatorname{ch}_2(\mathcal{U}')$ is divisible by $r$, but generates the second cohomology together with $c_1(\mathcal{U}'_x)$. Therefore, the latter must be coprime to $r$, in particular this is true for $c_1(\mathcal{U})$.
