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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

  1. $\mathcal{O}(k)$ over $\mathbb{P}^n$
  2. $\mathcal{O}(k)\oplus \mathcal{O}(l)$ 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 $\mathbb{P}^n$.

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  • $\begingroup$ First, you write $E^{\otimes 2}$ above, which is not correct, it should be $S^2 E$. Second, what exactly do you mean by `compute' in your question? What information do you want to extract other than saying it is the variety associated to the symmetric algebra? $\endgroup$ – Mohan Jul 19 '17 at 22:55
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    $\begingroup$ I want to compute a presentation of the vector bundle using relative spec. For example, $\underline{\text{Spec}}_{\mathcal{O}_X}(\mathcal{O}_X[y_1,y_2]) = \mathbb{A}^2_{X}$. $\endgroup$ – 54321user Jul 19 '17 at 23:02
  • $\begingroup$ Also, $\underline{\text{Spec}}_{\mathcal{O}_X}(\text{Sym}^\bullet(\mathcal{O}_X\oplus\mathcal{O}_X)) = \mathbb{A}^2_X$ $\endgroup$ – 54321user Jul 19 '17 at 23:25
  • $\begingroup$ I suggest that you try these for $n=1$ and already see that they are not so pleasant to write down (though it can be). For trivial bundles like $\mathcal{O}_X^n$, these are of course easy. $\endgroup$ – Mohan Jul 20 '17 at 2:04
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    $\begingroup$ For $k\geq 0$, theoretically it can be written down, but what do you wish to understand from such a complicated description in general? $\endgroup$ – Mohan Jul 20 '17 at 2:45

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