I am reading a book about the history of algebraic geometry.
In this I came upon an interesting definition of a very ample divisor:
If $X$ is a smooth projective variety, a divisor $D$ on $X$ is called very ample if it is the section of an immersion of $X$ in a projective space $\mathbb{P}^r$ with a hyperplane of $\mathbb{P}^r$ not containing $X$.
I've been using Hartshorne's definition that a very ample line bundle (divisor) is one which induces a closed embedding and such that the pullback of standard twisting sheaf is isomorphic to the line bundle.
How are these two definitions related? I don't see how the hyperplane from the first definition has anything to do with the definition in Hartshorne.
Best Answer
If $X$ is not too bad (I believe normal should be enough) you have an equivalence between Cartier divisor, line bundle and invertible sheaf. A line bundle $L$ gives you a Cartier divisor (up to linear equivalence) as the zero set of a section $s : X \to L$.
(Edit : as Mohan said, one should really put Cartier here : in fact, the sheaf $O_X(D)$ is an invertible sheaf, i.e a line bundle if and only if $D$ is Cartier)
Now, in term of $\mathcal O(1)$ on $\Bbb P^n$, a section $s$ of this bundle is by definition an homogenous polynomial of degree $1$, and its zero set will be an hyperplane $H$. Now, the pullback of $\mathcal O(1)$ to $X$ is $\mathcal O(1)_{|X}$ and of course $Z(s_{|D}) = Z(s) \cap D = D \cap H$. This shows why the definition of Hartshorne also gives you an hyperplane section.