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I am working with jet bundles on compact Riemann surfaces. So if we have a line bundle $L$ on a compact Riemann surface $C$ we can associate to it the $r$-th jet bundle $J^rL$ on $C$, which is a bundle of rank $r+1$. If we have a section $s\in H^0(C,L)$ then there is an induced section $D^rs\in H^0(C,J^rL)$ which is defined, locally on an open subset $U\subset C$ trivializing both $L$ and $\omega_C$, as the $(r+1)$-tuple $(f,f',\dots,f^{(r)})$, where $f\in O_C(U)$ represents $s$ on $U$.

Question 1. Does every section of $J^rL$ come from some $s\in H^0(C,L)$ this way?

Question 2. Do you know of any reference for a general description of the transition matrices attached to $J^rL$? I only know them for $r=1$ up to now and I am working on $r=2$.

Thank you in advance.

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This is rather old so maybe you figured out the answers already. Answer to Q1 is No. Not every global section of $J^r L$ comes from the "prolongation" of a section of $L$, not even locally. Consider for example the section in $J^1(\mathcal{O}_\mathbb{C})$ given in coordinates by $(0,1)$ (constant sections $0$ and $1$). This is obviously not of the form $(f,f')$.

The second question: maybe you find the explicit formulas for the transition of charts in Saunders "Geometry of Jet Bundle".

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    $\begingroup$ Thanks for your answer, it is never too late! I also found out later that $H^0(J^rL)$ is a direct sum of the vector spaces $H^0(L\otimes \omega_C^i)$ for $0\leq i\leq r$, so the first one of these is $H^0(L)$. And thanks for the book reference! $\endgroup$
    – Brenin
    Nov 18, 2013 at 21:23
  • $\begingroup$ @Brenin What's $\omega^{i}_C$ here? Isn't it supposed to be that $J^r L \simeq Sym^{\leq r}(\mathcal{T}^*_C)$? $\endgroup$ Sep 6, 2016 at 11:26

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