# Geometric Interpretation of Automorphisms of Projective Bundles

Let $$\mathcal{E}$$ be a rank three vector bundle on $$\mathbb{P}^1$$. It splits as $$\mathcal{E}\cong\mathcal{O}(a_1)\oplus\mathcal{O}(a_2)\oplus\mathcal{O}(a_3)$$. It's not hard to see that any automorphism of $$\mathbb{P}(\mathcal{E})$$ acts trivially on the Picard group, which is generated by the class $$f$$ of a fiber and the class $$h$$ of $$\mathcal{O}_{\mathbb{P}(\mathcal{E})}(1)$$. In particular, because any automorphism takes fibers to fibers, we obtain a surjective map $$\text{Aut}(\mathbb{P}(\mathcal{E}))\rightarrow \text{Aut}(\mathbb{P}^1)$$. We get an exact sequence $$1\rightarrow G\rightarrow \text{Aut}(\mathbb{P}(\mathcal{E}))\rightarrow \text{Aut}(\mathbb{P}^1)\rightarrow 1$$ We think of $$G$$ as automorphisms that fix each fiber as a set and leave $$\mathbb{P}^1$$ fixed. Algebraically, we can see $$G=\mathbb{P}\text{Aut}(\mathcal{E})$$. We have $$\text{Aut}(\mathcal{E})\subset \text{End}(\mathcal{E})=H^0(\mathbb{P}^1,\mathcal{E}\otimes \mathcal{E}^{\vee})$$. We can view an element of $$H^0(\mathbb{P}^1,\mathcal{E}\otimes \mathcal{E}^{\vee})$$ as a $$3\times 3$$ matrix with entries $$a_{ij}\in H^0(\mathbb{P}^1,\mathcal{O}(a_i-a_j))$$. I'd like to get my hands on what these automorphisms do in a geometric sense. It may be instructive to consider a specific example. Let $$\mathcal{E}=\mathcal{O}(3)\oplus\mathcal{O}(3)\oplus\mathcal{O}(4)$$. Our matrix is $$\begin{pmatrix} a & b & 0 \\ c & d & 0 \\ l_1 & l_2 & e \end{pmatrix}$$ where $$a,b,c,d,e$$ are scalars and $$l_1, l_2\in H^0(\mathbb{P}^1,\mathcal{O}(1))$$. The condition that this matrix defines an automorphism is that the determinant doesn't vanish, which just means that $$(ad-bc)e\neq 0$$. After projectivizing, we can set $$e=1$$. Intuitively, this matrix breaks up in to a few parts. The top left $$2\times 2$$ block seems to be a $$GL_2$$. The bottom left part is a $$H^0(\mathcal{O}(1))^{\oplus 2}=\mathbb{A}^4$$. How can I interpret these parts of the matrix geometrically? What types of automorphisms do they correspond to? One idea I have is the following. There is a natural divisor $$\mathbb{P}(\mathcal{O}(3)\oplus\mathcal{O}(4))$$ inside $$\mathbb{P}(\mathcal{E})$$ given by one of the obvious inclusions. Using the well-known Chow ring of $$\mathbb{P}(\mathcal{E})$$, I was able to compute that this divisor lies in class $$h-3f$$. Using the projection formula, we see that $$h^0(\mathbb{P}(\mathcal{E}),\mathcal{O}(h-3f))=4$$. From here, it seems like the bottom left part of the matrix moves around divisors in the class $$h-3f$$. Is this a reasonable interpretation? What about the $$GL_2$$ block? Perhaps it corresponds in some way to the divisor $$\mathbb{P}(\mathcal{O}(3)\oplus\mathcal{O}(3))$$?