M. Atiyah in "VECTOR BUNDLES OVER AN ELLIPTIC CURVE" defined ample line bundle $E$ on $X$ as satisfying the following conditions:


1. Canonical map $H^0(X, E)\to E\_x$ is surjective for any $x\in X$.
2. $H^q(X,E)=0$ for $q>0$.


But in standard textbooks like Hartshorne there is another definition:


$E^{\otimes n}$ is very ample for some $n\in \mathbb N$.


Are these definitions really equivalent? Where can I find proof of this?



Equivalent definitions of ample bundles
Even a very ample line bundle does not need to be ample in the sense of the first definition (consider $\mathcal O(1)$ on a smooth plane curve of degree $>4$ or, more generally, on any smooth hypersurface of high enough degree). I do not know whether ampleness in the sense of the 1st definition implies ampleness in the sense of teh second definition.


Nowadays the standard definition of ampleness is as given in Hartshorne's book.


