Let $X$ smooth projective curve over $\mathbb{C}$, fix a line bundle $L$ of degree $d$, and let $\mathcal{M}(r,d)$ denote the moduli space of semistable vector bundles of rank $r$ and degree $d$. It is known to be a normal projective variety.

There is a determinant map $det: \mathcal{M}(r,d)\rightarrow Pic^d(X)$, where $Pic^d(X)$ denotes the isomorphism class of line bundles of degree $d$.

Is it true that the $det$ map is a fibre bundle? I recently heard this statement, but I can't seem to prove it. Can someone give some idea, or give some reference if possible? Thanks in advance!

  • 4
    $\begingroup$ Fix a line bundle $L\in Pic^d(X)$, and consider the étale covering $\pi :Pic^0(X)\rightarrow Pic^d(X)$ given by $M\mapsto M^r\otimes L$. It is easy to see that the pull back of $\ \det\ $ under $\pi $ is the second projection $\mathcal{M}_L\otimes Pic^0(X)\rightarrow Pic^0(X)$. Thus $\ \det\ $ is an étale fiber bundle, and even more: it becomes trivial after a finite étale covering. $\endgroup$ – abx Dec 9 '19 at 19:33

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