I have this situation, $r$ and $s$ are positive integers: we define $\mathcal{E}$ to be the vector bundle on $\mathbb{P}^s$ whose associated locally free sheaf is $$ \mathcal{O}_{\mathbb{P}^s} \oplus \mathcal{O}_{\mathbb{P}^s}(1)^{r+1}. $$ I would like to describe explicitly $\mathcal{E}$, how can I do it step-by-step?
I'm putting myself in the situation where $r=1$ and $s=2$, so $\mathcal{E}$ is the vector bundle associated to the sheaf $\mathcal{O}_{\mathbb{P}^2} \oplus \mathcal{O}_{\mathbb{P}^2}(1)^{2}$.
- Clearly $\mathcal{O}_{\mathbb{P}^2}$ are the regular functions on $\mathbb{P}^2$, hence $\mathbb{C}$.
- $\mathcal{O}_{\mathbb{P}^2}(1)$ can be associated to the homogeneous polynomials of degree 1, hence $\mathbb{C}[x_0,x_1,x_2]_1$, so $\mathcal{O}_{\mathbb{P}^2}(1)^2=\mathcal{O}_{\mathbb{P}^2}(2)$ (am I right?) is associated to the homogeneous polynomials of degree 2, hence $\mathbb{C}[x_0,x_1,x_2]_2$.
Since $\dim \mathbb{C}[x_0,x_1,x_2]_2=6$ as a vector space, I expect that $\mathcal{E}$ is something locally as $U_i \times \mathbb{C}^7$ where $U_i=\{x_i \ne 0\}$ is the usual covering of $\mathbb{P}^2$. Am I right? But, how $\mathcal{E}$ looks globally?