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By Riemann-Roch, for a degree 1 line bundle on a genus 2 Riemann surface the space of global holomorphic sections has dimension between $0$ and $2$. Can someone show an explicit example of a degree 1 line bundle without global (non identically vanishing) sections? (or a proof/reference that a degree 1 line bundle on a genus 2 surfaces always has a non trivial global section)

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  • $\begingroup$ related: math.stackexchange.com/q/1193907/22002 $\endgroup$ – Lor Mar 17 '15 at 14:02
  • $\begingroup$ May be i am missing something but can you explain how you get the bounds on space of holom sections from RR? $\endgroup$ – DBS Mar 17 '15 at 15:53
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    $\begingroup$ This is in Gunning, Lectures on Riemann Surfaces, chapter 7.b, in the discussion after Theorem 13. $\endgroup$ – Lor Mar 17 '15 at 16:41

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