For Projective bundle $\pi:\mathbb P(\mathcal E)\rightarrow X$, what's the natural morphism $\pi^*\mathcal E\rightarrow \mathcal O(1)$?

Question

I'm reading Hartshorne proposition 7.11.

Proposition 7.11.b Let $X$ be a Notherian scheme and $\mathcal E$ be a locally free sheaf. Then there is a natural subjective morphism $\pi^*\mathcal E\rightarrow \mathcal O(1)$ where $\pi:\operatorname{Proj}(\operatorname{Sym}\mathcal E)\rightarrow X$.

My questions:

1. How to construct the natural morphism $\pi^*\mathcal E\rightarrow \mathcal O(1)$? Why it is surjective?

2. In the case $\operatorname{Proj}(A[x_0,\dots,x_n])\rightarrow \operatorname{Spec}(A)$, we know $\mathcal E = A[x_0,\dots,x_n]^\sim$. Then, what is $\pi^*\mathcal E$ and what is the natural morphism $\pi^*\mathcal E\rightarrow \mathcal O(1)$ in this special case?

Answer 1

Question: 1. How to construct the natural morphism $\pi^*\mathcal E\rightarrow \mathcal O(1)$? Why it is surjective? 2. In the case $\operatorname{Proj}(A[x_0,\dots,x_n])\rightarrow \operatorname{Spec}(A)$, we know $\mathcal E = A[x_0,\dots,x_n]^\sim$. Then, what is $\pi^*\mathcal E$ and what is the natural morphism $\pi^*\mathcal E\rightarrow \mathcal O(1)$ in this special case?

Answer: Locally $E:=A\{e_0,..,e_n\}$ is a free $A$-module of rank $n+1$ and we let $E^*:=E\{x_0,..,x_n\}$ and $Sym^*_A(E^*) \cong A[x_0,..,x_n]$ with $\pi:\mathbb{P}(E^*):=Proj(Sym^*_A(E^*))\rightarrow S:=Spec(A)$. it follows there is a canonical surjective map

$$\gamma: A[x_0,..,x_n]\otimes E^* \rightarrow A[x_0,..,x_n](1)$$

defined by

$$\gamma(\sum_i f_i(x)\otimes x_i):= \sum_i f_i(x)x_i.$$

This gives a surjective map

$$\gamma^*: \pi^*(E^*) \rightarrow \mathcal{O}(1) \rightarrow 0.$$

$\gamma^*$ is surjective since $\gamma$ is surjective. This construction has the property that $\pi_*\mathcal{O}(l) \cong Sym^l_A(E^*)$ for $l \geq 1$. Hence $H^0(\mathbb{P}(E^*), \mathcal{O}(1)) \cong E^*$. There is an action of $GL_A(n+1)$ on $E$ inducing an action on $E^*$ and $Sym^l(E^*)$ hence you get canonical actions of $GL_A(n+1)$ on $H^0(\mathbb{P}(E^*), \mathcal{O}(l))$ for any $l \geq 1.$

Note: If instead you define $\mathbb{P}(E):=Proj(Sym^*_A(E))$ you must use $E$ instead of $E^*$. It is more natural to use $E^*$: The functions $x_i:=e_i^*$ are "coordinate functions" or "dual coordinates".

Example: As an example: If $V:=k\{e_0,..,e_n\}$ and $V^*:=k\{x_0,..,x_n\}$ it follows $\mathbb{P}(V^*):=Proj(Sym_k^*(V^*)) \cong Proj(k[x_0,..,x_n])$ and $\mathbb{P}(V) \cong Proj(k[e_0,..,e_n])$. It is more natural to consider the dual space $V^*$ and the coordinate functions $x_i$.