# Line bundle of degree 1 on a genus 2 surface without global holomorphic sections

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)

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