The Stacks project

Remark 27.14.2. Assumptions as in Lemma 27.14.1 above. The image of the morphism $r_{\mathcal{L}, \psi }$ need not be contained in the locus where the sheaf $\mathcal{O}_ X(1)$ is invertible. Here is an example. Let $k$ be a field. Let $S = k[A, B, C]$ graded by $\deg (A) = 1$, $\deg (B) = 2$, $\deg (C) = 3$. Set $X = \text{Proj}(S)$. Let $T = \mathbf{P}^2_ k = \text{Proj}(k[X_0, X_1, X_2])$. Recall that $\mathcal{L} = \mathcal{O}_ T(1)$ is invertible and that $\mathcal{O}_ T(n) = \mathcal{L}^{\otimes n}$. Consider the composition $\psi $ of the maps

\[ S \to k[X_0, X_1, X_2] \to \Gamma _*(T, \mathcal{L}). \]

Here the first map is $A \mapsto X_0$, $B \mapsto X_1^2$, $C \mapsto X_2^3$ and the second map is ( By the lemma this corresponds to a morphism $r_{\mathcal{L}, \psi } : T \to X = \text{Proj}(S)$ which is easily seen to be surjective. On the other hand, in Remark 27.9.2 we showed that the sheaf $\mathcal{O}_ X(1)$ is not invertible at all points of $X$.

Comments (2)

Comment #5005 by Laurent Moret-Bailly on

The image of should be , not .

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Your email address will not be published.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 01NL. Beware of the difference between the letter 'O' and the digit '0'.