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Let $X$ be a smooth projective curve of genus $g \geq 2$ over $\mathbb{C}.$ Does there exist a line bundle $L$ on $X$ of degree $\deg L= 2g-1$ such that it is generated by global sections?

(One can show that if $L$ is a line bundle of degree $\deg L=2g-1$ such that the canonical sheaf $\omega_{X}$ is not a subsheaf of $L$, then L is generated by global sections. So it is enough to show the existence of such a line bundle. I do not know whether this is easy? )

fish_monster
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Suhas
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  • Stupid comment: if $X = \mathbf{P}^1$ (i.e. $g=0$) then $L = \mathscr{O}(-1)$ and then global generation is impossible. – Hoot Sep 24 '15 at 13:29
  • @Suhas: if you are still interested to see why a general line bundle of degree at most g-1 has no sections see the proof of proposition 5.12 in Eisenbud's book: The Geometry of Syzygies: A second course in Commutative Algebra and Algebraic Geometry – adrido Oct 01 '15 at 04:08

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As Hoot pointed out, this is false for $g=0$, but also false for $g=1$. On the other hand, if $g\geq 2$, this is true. Basic fact to be noticed is that for any $g$, a general line bundle of degree at most $g-1$ has no non-zero sections. So, if $g\geq 2$, a general line bundle $L$ of degree one has no non-zero sections and thus $\omega_X\otimes L$ is a degree $2g-1$ line bundle with the property you seek.

Mohan
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  • Thanks Mohan. But I am not able to see why for any g, a general line bundle of degree utmost g-1 has no non-zero sections. Will you please let me know the reason for this fact or tell some reference. – Suhas Sep 24 '15 at 15:39
  • You can look up any book, for example Hartshorne's chapter on Curves. I had given a proof in stackexchange recently, may be you can look it up. – Mohan Sep 24 '15 at 16:11
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    @Mohan: Will you please send the link, where you have answered the above fact – user177523 Sep 25 '15 at 06:52