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I am studying Castelnuovo's rationality criterion for surfaces.

Let $S$ be a projective surface and $K$ a canonical divisor on $S$. Let's use the notation $h^i(S,\mathcal{O}_S)=dimH^i(S,\mathcal{O}_S)$.

The condition $P_2=h^0(S,\mathcal{O}_S(2K))=0$ implies $p_g(S)=h^2(S,\mathcal{O}_S)=h^0(S,\mathcal{O}_S(K))=0$.

Is this correct?

I can't find a proof of this.

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    $\begingroup$ If a line bundle $L$ has a non zero section $s$, then $s\otimes s$ is a non-zero section of $L^{\otimes 2}$. $\endgroup$
    – Henri
    Commented Jan 21, 2014 at 21:40
  • $\begingroup$ @Henri this also means that if $P_n=0$ for some $n$, then $p_g=0$, right? $\endgroup$
    – idioteca
    Commented Jan 22, 2014 at 10:09
  • $\begingroup$ Yes, exactly. Take $s ^{\otimes n}$ this time, and you get the same thing. $\endgroup$
    – Henri
    Commented Jan 22, 2014 at 10:16

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