# 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?

• $c_1(\mathcal{U}_x)=c_1(L)=\deg L$. – Chen Jiang Dec 9 '17 at 1:42
• How do you see that? – Z.A.Z.Z Dec 9 '17 at 12:38
• This is just by definition. $\mathcal{U}_x$ is just one of the semistable vector bundle in the moduli. – Chen Jiang Dec 9 '17 at 17:57
• I don't see that! $\mathcal U_x$ is a bundle over the moduli space. If $e\in M$ then I agree that $\mathcal U_e$ is a bundle as you said?! – Z.A.Z.Z Dec 10 '17 at 14:00
• Oh, you are right. I thought you are asking about the fiber over $M$. Sorry. – Chen Jiang Dec 11 '17 at 17:24