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my question concerns a very ample line bundle $L$ on a projective k-scheme. It gives a closed immersion

$i:X \rightarrow P^{N-1}_{k}$

to some projective space over k.

It can be viewed as induced by global sections $s_ 1,...,s_N$ of $L$.

Why do people always say that these sections form a basis of $H^{o}(X,L)$? Surely they generate $L$ in the sense of: generating in every stalk. But why is $N=dim(H^{o}(X,L)$?

Thanks a lot!

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    Being globally generated corresponds to a surjection of sheaves $\pi:\mathcal{O}_X^{\oplus N}\twoheadrightarrow L$. Being surjective on global sections corresponds to vanishing higher cohomology, so the sections $s_1,\ldots,s_N$ will generate $L(X)$ if and only if $H^1(\ker(\pi))=0$ if and only if $H^1(\mathcal{O}_X^{\oplus N})\hookrightarrow H^1(L)$.2011-08-19

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