Given a coherent sheaf $\mathcal{E} \in \text{Coh}(X)$ over a scheme there is a way to associate a relatively affine scheme over $X$. This is done by constructing the $\mathcal{O}_X$-algebra $$ \text{Sym}^\bullet(\mathcal{E}) = \mathcal{O}_X \oplus \mathcal{E} \oplus \text{Sym}^2(\mathcal{E})\oplus \text{Sym}^3(\mathcal{E})\oplus \cdots $$ and then taking relative spec $$ \mathbb{V}(\mathcal{E}) = \underline{\text{Spec}}_{\mathcal{O}_X}(\text{Sym}^\bullet(\mathcal{E})) $$ I am having trouble figuring how to compute basic examples of this, so
how can I compute $\mathbb{V}(\mathcal{E})$ in simple cases?
such as
- $\mathcal{O}(k)$ over $\mathbb{P}^n$
- $\mathcal{O}(k)\oplus \mathcal{O}(l)$ over the same space
I know that I can use these computations to find the same associated vector bundles to some projective variety using pullbacks.
As per Ben's suggestion, I'll look at $\mathbb{P}^1$. Since we have the embedding $$ \mathcal{O}(-1) \xrightarrow{\begin{bmatrix} x \\ y \end{bmatrix}} \mathcal{O}\oplus\mathcal{O} $$ I expect this to be the sub-variety of $\mathbb{A}^2_{\mathbb{P}^1}$ defined by the equation $$ \frac{\mathcal{O}_{\mathbb{P}^1}[a,b]}{(ya - xb)} $$ where $\mathbb{P}^1 = \text{Proj}(\mathbb{C}[x,y])$. In general, this should be given by all of the linear relations cutting out a line over each point of $\mathbb{P}^n$.