# Definition of very ample line bundle.

I am reading Vakil's algebraic geometry. He gives a definition of very ample line bundle as followings:

Suppose $$\pi: X \rightarrow \operatorname{Spec} A$$ is a proper morphism, and $$\mathscr{L}$$ is an invertible sheaf on X. Then we say $$\mathscr{L}$$ is very ample over $$A$$ if $$X \cong$$ Proj $$S_{\bullet}$$ where $$S_{\bullet}$$ is a finitely generated graded ring over $$A$$ generated in degree 1 and $$\mathscr{L} \cong \mathscr{O}_{\text {Proj} S_{\bullet}}(1)$$.

Then there is one exercise

Show that $$\mathscr{L}$$ is very ample if and only if there exist a finite number of global sections $$s_{0}, \ldots s_{n}$$ of $$\mathscr{L},$$ with no common zeros, such that the morphism $$\left[\mathrm{s}_{0}, \ldots, \mathrm{s}_{\mathrm{n}}\right]: \mathrm{X} \rightarrow \mathbb{P}_{\mathrm{A}}^{\mathrm{n}}$$is a closed embedding.

However, I think the exercise is wrong if we use his definition. For example, considering the closed embedding $$\mathbb{P}^{1} \rightarrow \mathbb{P}^{2}$$ given by $$[x: y] \mapsto\left[x^{2}: x y: y^{2}\right]$$ it pulls $$\mathcal{O}_{\mathbb{P}^{2}}(1) \text { to } \mathcal{O}_{\mathbb{P}^{1}}(2)$$. Obviously, $$\mathcal{O}_{\mathbb{P}^{1}}(2)$$ satisfies the condition in the exercise, but it is not isomorphic to $$\mathcal{O}_{\mathbb{P}^{1}}(1)$$.

So I think the definition of very ample line bundle is not proper. And actually we should use the exercise as the definition.

Am I right? If not, could you tell me the exact definition of very ample line bundle? Thank you very much.

• I've edited your post to remove a \text tag inside the latex formatting that was causing a part of your question to be unreadable. Please just leave the latex environment for this in the future. – KReiser Nov 7 '19 at 21:05

The definition is fine - it's an existence statement, not a uniqueness statement. To fix your issue, we write $$\Bbb P^1_k$$ as $$\operatorname{Proj} k[x^2,xy,y^2]$$ instead of $$\operatorname{Proj} k[x,y]$$. As $$\operatorname{Proj} k[x^2,xy,y^2]$$ has $$\mathcal{O}(1)=k\langle x^2,xy,y^2\rangle$$, we see that the definition is satisfied.
This generalizes: for any graded ring $$R$$, the graded ring $$R^{(d)}$$ defined by $$R^{(d)}=\bigoplus_{n\in\Bbb Z_{\geq 0}} R_{dn}$$ has the same $$\operatorname{Proj}$$ as $$R$$. So if $$\mathcal{O}_{\operatorname{Proj} R}(1)$$ is very ample, then $$\mathcal{O}_{\operatorname{Proj} R}(d)$$ is also very ample for any $$d>0$$.
• By the way, I think you want to say that if $\mathcal{O}_{\mathrm{Proj} R^{(d)}}$$(1) is very ample, then \mathcal{O}_{\mathrm{Proj} R}$$ (d)$ is very ample? – Mike Nov 7 '19 at 21:47