Interpretation of $Hom_{\mathcal{O}_X} (\mathcal{E}, \mathcal{F} ) \cong \mathcal{E}^* \otimes \mathcal{F}$

Question

Let $\mathcal{E}$ be a locally free sheaf of $\mathcal{O}_X$-modules of finite rank. It is a standard exercise to show that for any $\mathcal{O}_X$-module $\mathcal{F}$, $$Hom_{\mathcal{O}_X} (\mathcal{E}, \mathcal{F} ) \cong \mathcal{E}^* \otimes_{\mathcal{O}_X} \mathcal{F} $$ where $\mathcal{E}^*$ is the sheaf $Hom_{\mathcal{O}_X}(\mathcal{E}, \mathcal{O}_X)$.

Let $X=\mathbb{P}^1=\operatorname{Proj} k[X_0,X_1]$ and $\mathcal{E} \cong \mathcal{O}(-1)$ and let $\mathcal{F} \cong \mathcal{O}(2)$.

Then $\operatorname{Hom}_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F}) \cong \Gamma(X, \mathcal{O}(3))$ where $ \Gamma(X, \mathcal{O}(3)) $ is generated as a $\Gamma(X, \mathcal{O}_X)= k$-module by $X_0^3, \ X_0^2X_1, \ X_0 X_1^2, \ X_1^3$. This means that there exist $4$ disinct morphism $\mathcal{O}(-1) \to \mathcal{O}(2)$ up to $\Gamma(X, \mathcal{O}_X^*)$.

I know each of the generators $X_0^3, \ X_0^2X_1, \ X_0 X_1^2, \ X_1^3$ corresponds to a morphism $\phi \in \operatorname{Hom}(\mathcal{O}(-1), \mathcal{O}(2))$.

Suppose $\varpi \in \operatorname{Hom}(\mathcal{O}(-1), \mathcal{O}(2))$ corresponding to the generator $X_0^3$. How is $X_0^3$ related to the image of $\mathcal{O}(-1)$ under $\varpi$ in $\mathcal{O}(2)$ ?

At some point I was thinking that $X_0^3$ would generate (over k) all global sections of the image of $\mathcal{O}(-1)$ as a subsheaf ( I guess here I would be imagining that $\varpi$ is an inclusion) of $\mathcal{O}(2)$

Answer 1

$\mathcal O(-1)$ doesn't actually have any global sections, but it has lots of local sections. For example, $s = (X_0 + 4X_1) / X_1^2$ is a local section of $\mathcal O(-1)$, which is defined on the open set $\mathbb P^1 \setminus V(X_1)$. Under the action of $\varpi = X_0^3 \in {\rm Hom}(\mathcal O(-1), \mathcal O(2))$, $s$ maps to $\varpi(s) = X_0^3(X_0 + 4X_1) / X_1^2$, which is a local section of $\mathcal O(2)$ defined on the same open set.

It is true that the sheaf morphism $\varpi : \mathcal O(-1) \to \mathcal O(2)$ is injective on all stalks. The sheaf morphism $\varpi$ is also surjective on all stalks except for the stalk at $p := [0 : 1]$, and the cokernel of $\varpi$ is a skyscraper sheaf of rank $3$ supported at $p$.