Timeline for Fiber of the Hitchin map
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Oct 15, 2020 at 12:15 | comment | added | Sam Gunningham | Moreover, for any open affine $U\subseteq C$, we have that $\Gamma(U;E) = \Gamma(\pi^{-1}(U);L_E)$ injects in to the generic stalk $E_{\eta_C} = (L_E)_{\eta_{C_a}}$ (as $E$ is torsion free). Thus $L_E$ must be torsion free. I hope this makes sense! | |
| Oct 15, 2020 at 12:13 | comment | added | Sam Gunningham | The condition that $C_a$ is irreducible means that we have a degree $n$ extension of function fields $K(C) \hookrightarrow K(C_a)$. Then $L_E$ restricted to the generic point must be a rank $1$ $K(C_a)$-vector space, as we know that it is rank $n$ over $K(C)$. Thus $L_E$ is generically rank 1. | |
| Oct 14, 2020 at 14:48 | comment | added | Aoki | Thanks, your comment is very helpful. I read the paper of BNR, but I don't understand the reason $L_E$ is torsion-free of rank $1$. | |
| Oct 14, 2020 at 14:27 | comment | added | Sam Gunningham | There are details in Section 3 of this paper math.unice.fr/~beauvill/pubs/bnr.pdf | |
| Oct 14, 2020 at 14:27 | comment | added | Sam Gunningham | Note that in general (for an affine morphism $\pi$) the functor $\pi_\ast$ induces an equivalence of categories between quasi-coherent sheaves on $C_s$ andquasi-coherent $\pi_\ast(\mathcal O_{C_s})$-modules on $C$. Thus $E$ corresponds to a quasi-coherent sheaf $L_E$ on $C_s$ with $\pi_\ast (L_E) \cong E$. It remains to show that $L_E$ is an invertible sheaf. You can do this by first showing that $L_E$ must be torsion-free of rank 1. | |
| Oct 14, 2020 at 13:57 | history | asked | Aoki | CC BY-SA 4.0 |