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Let $C$ a compact riemann surface of positive genus and $\omega_C$ the canonical divisor over $C$ with standard degree $2g-2$. Take on $C$ a divisor of positive degree $d$ and set $$V=H^0(C,\omega_C(D))$$ the set of section of $\omega$ with poles on $D$.
If i take an element in $V$ it has degree $2g-2+d$ but can i see an element of $V$ as a meromorphic section of $\omega_C$? And in this case must its degree be zero?

I'm confused about two both interpretation.

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  • $\begingroup$ @Ted Shifrin now is clear. There is another question that i cannot solve. If a take the space $H^0(C,\omega_C(2D)/ \omega_C(-2D))$ can i see an element of this as a rational section of $\omega_C$ around $D$ with poles in $D$ so an element of $H^0(C,\omega_C(2D)/ \omega_C)$? $\endgroup$ – dario May 15 '15 at 15:18
  • $\begingroup$ I don't understand this notation. Do you mean $H^0(C,\omega_C(2D))/H^0(C,\omega_C(-2D))$? $\endgroup$ – Ted Shifrin May 15 '15 at 16:32
  • $\begingroup$ @Ted Shifrin With the notation $H^0(C,\omega_C(2D)/ \omega_C(-2D))$ i mean the vector space of all section of $\omega_C$ with poles on $D$ modulo those vanishing on $D$. $\endgroup$ – dario May 15 '15 at 17:51
  • $\begingroup$ @Ted Shifrin If you want see also this question that is related link $\endgroup$ – dario May 15 '15 at 17:56
  • $\begingroup$ @Ted Shifrin Here the article that i'm reading where the notation appear. At the end of page 615 and at the beginning of page 616 if you need. link $\endgroup$ – dario May 15 '15 at 18:04

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