The Stacks project

Example 29.38.4. Let $k$ be a field. Consider the graded $k$-algebra

\[ A = k[U, V, Z_1, Z_2, Z_3, \ldots ]/I \quad \text{with} \quad I = (U^2 - Z_1^2, U^4 - Z_2^2, U^6 - Z_3^2, \ldots ) \]

with grading given by $\deg (U) = \deg (V) = \deg (Z_1) = 1$ and $\deg (Z_ d) = d$. Note that $X = \text{Proj}(A)$ is covered by $D_{+}(U)$ and $D_{+}(V)$. Hence the sheaves $\mathcal{O}_ X(n)$ are all invertible and isomorphic to $\mathcal{O}_ X(1)^{\otimes n}$. In particular $\mathcal{O}_ X(1)$ is ample and $f$-ample for the morphism $f : X \to \mathop{\mathrm{Spec}}(k)$. We claim that no power of $\mathcal{O}_ X(1)$ is $f$-relatively very ample. Namely, it is easy to see that $\Gamma (X, \mathcal{O}_ X(n))$ is the degree $n$ summand of the algebra $A$. Hence if $\mathcal{O}_ X(n)$ were very ample, then $X$ would be a closed subscheme of a projective space over $k$ and hence of finite type over $k$. On the other hand $D_{+}(V)$ is the spectrum of $k[t, t_1, t_2, \ldots ]/(t^2 - t_1^2, t^4 - t_2^2, t^6 - t_3^2, \ldots )$ which is not of finite type over $k$.


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01VO. Beware of the difference between the letter 'O' and the digit '0'.