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M. Atiyah in "VECTOR BUNDLES OVER AN ELLIPTIC CURVE" defined ample line bundle $E$ on $X$ as satisfying the following conditions:
Canonical map $H^0(X, E)\to E_x$ is surjective for any $x\in X$.
$H^q(X,E)=0$ 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?
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.
$\mathcal O_x$ for each $x\in X$"). If $X$ is proper and positive-dimensional (and, say, connected), this condition does not hold: a local ring of $X$ cannot be generated by constants only!
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– Serge Lvovski
Apr 19 '13 at 13:05
Let $X$ be an elliptic curve.
Then a divisor $E$ which is ample in the sense of the first definition is necessarily ample in the usual sense. Conversely, if it is ample in the usual sense and $\deg E \geq 2$ then it is ample in the sense of the first definition.
In fact, assume that $E$ ample in the sense of the first definition. Then $(2)$ implies that $\mathcal{O}_X(E) \neq \mathcal{O}_X$ and $(1)$ implies that $\dim H^0(X, E) \geq 2$, because a non-trivial divisor with $\dim H^0(X, E) =1 $ on an elliptic curve cannot be globally generated. Then by $(2)$ and Riemann-Roch it follows $$\deg E = \dim H^0(X, E) \geq 2,$$ hence $E$ is ample in the usual sense by [Hartshorne, Algebraic Geometry, Corollary 3.3 page 308].
Conversely, assume that $E$ is ample in the usual sense and $\deg E \geq 2$. This last condition implies that $|E|$ is base-point free on $X$, hence $E$ is globally generated and $(1)$ holds. Moreover, since $\omega_X = \mathcal{O}_X$, by Kodaira Vanishing Theorem and Serre duality one deduces $(2)$. So $E$ is ample in the sense of the first definition.
