# Automorphism of $k$ varieties is induced by automorphism of $\mathbb{P}_k^n$ (exercise 5.1.24 in Q Liu)

The exercise is given below.

$$5.1.24.$$ Let $$X$$ be an algebraic variety over a field $$k$$. Let $$f : X \to Y := \mathbb{P}^{n}_{k}$$ be a morphism of algebraic $$k-$$varieties. Let $$\mathcal{L} = f^{*}\mathcal{O}_{Y}(1)$$. Let $$\sigma$$ be an automorphism of $$X$$ such that $$\sigma^{*}\mathcal{L} \simeq \mathcal{L}$$. Show that there exists an automorphism $$\sigma'$$ of $$Y$$ such that $$\sigma' \circ f = f \circ \sigma$$.

From what I know, a $$k-$$ automorphism of $$\mathbb{P}^{n}_{k}$$ comes from an automorphism of the graded $$k-$$ algebra $$k[X_{1}, ..., X_{n}]$$. With the hypothesis, if $$g := f \circ \sigma$$ we have $$f^{*}\mathcal{O}_{Y}(1) = g^{*} \mathcal{O}_{Y}(1)$$. From here I really don't know what to do. Any help would be appreciated.

• Please don't use images for questions because such questions don't show up in search results. Jul 9, 2023 at 8:26
• Hi, you're right : I edit my post. Jul 10, 2023 at 7:54
• The annoying thing for me is that the construction of $f^{*}$ is not explicit. Jul 13, 2023 at 8:44

Let $$V = H^0(X, \mathcal L)$$. Then the map $$f$$ corresponds to the quotient $$V \otimes \mathcal O_X \to \mathcal L \to 0,$$ so we can think of $$f$$ as a map $$f: X \to \mathbb P(V)$$¹.
Now choose an isomorphism $$\varphi: \sigma^* \mathcal L \to \mathcal L$$. This induces an isomorphism on global sections $$\phi: V \to V$$, which in turn induces an isomorphism $$\sigma': \mathbb P(V) \to \mathbb P(V)$$.
Now it remains to check that $$\sigma'$$ is compatible with $$\sigma$$. Can you do this on your own?
¹ I'm using Grothendieck's version of $$\mathbb P(V)$$ here, i.e. a point in $$\mathbb P(V)$$ is an equivalence class of quotients $$V \to k \to 0$$. Two quotients are equivalent iff they have the same kernel. The map $$f$$ then maps each closed point $$x \in X$$ to the quotient $$[V \to \mathcal L_x \otimes_{\mathcal O_{X,x}} k(x) \to 0]$$.
• Thanks for your answer. I don't understand what is $\mathbb{P}(V)$. Do you have references for that? Jul 15, 2023 at 9:14
• @Analyse300 That is the projective space over $V$. If you learned that projective space $\mathbb P^n$ consists of lines through the origin $L \subset k^{n+1}$, then in my notation, $\mathbb P(V)$ parametrises lines in the vector space $V^*$. A lines $L \subset V^*$ is the same as a quotient $V \to L^* \to 0$. Mar 12 at 9:05