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

Here the first map is $A \mapsto X_0$, $B \mapsto X_1^2$, $C \mapsto X_2^3$ and the second map is (27.10.1.3). 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 $X$.