Timeline for Is the determinant map $det:\mathcal{M}(r,d)\rightarrow Pic^d(X)$ on moduli space of semistable vector bundles a fibration?
Current License: CC BY-SA 4.0
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| Dec 9, 2019 at 19:33 | comment | added | abx | 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. | |
| Dec 9, 2019 at 18:28 | history | asked | Hajime_Saito | CC BY-SA 4.0 |