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I understand that holomorphic square roots of the canonical line bundle of a compact Riemann surface always exist, and that there are $2^{2g}$ choices of such a root.

But what about further roots? When the genus of the Riemann surface is larger than $1$, we can ask for further roots, such as a square root of a square root of the canonical bundle.

Is this always possible? In other words, if $K$ is the canonical bundle, can I always keep taking roots so that eventually I have $K=L^{2g-2}$ for some degree $1$ holomorphic line bundle $L$?

Why or why not?

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    $\begingroup$ What's your proof for $K$ having a square root? I feel it probably only uses the fact that $K$ has even degree. In the end everything has to come from $0 \to \operatorname{Pic}^0X \to \operatorname{Pic} X \to \mathbb Z \to 0$. $\endgroup$ – Hoot Jun 10 '16 at 5:32

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