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In the Algebraic Geometry course I am following we have defined vector bundles in the following way: Given a locally free sheaf $\mathscr{E}$ on a scheme $S$ we can define a functor, $$\mathbb{V}(\mathscr{E}): \operatorname{Sch}/S \to \operatorname{Set}$$ $$(Y \xrightarrow{f} S) \mapsto f^{*}\mathscr{E}(Y)$$ The vector bundle over $S$ associated to $\mathscr{E}$ will be the representing object of this functor $\vert \mathscr{E} \vert \xrightarrow{\pi} S$. I had to show that for every $s \in \mathscr{E}(S)$ we can find a closed subscheme of $S$, $V(s)$ such that a morphism $Y \xrightarrow{f} S$ factorizes through $V(s)$ if and only if $f^{*}(s)=0$.

To do this I tried to define the "vanishing set" of the section using the set of points where the "coordinates" of $s$ lie in the maximal ideal but I had problems to finish the proof. However we can give a more geometrical interpretation:

$\mathbb{V}(\mathscr{E})(X \xrightarrow{id} X)=\mathscr{E}(X)$ therefore we can identify global sections of the sheaf with global sections of the bundle. Now we note that $f^{*}\mathscr{E}$ is also locally free therefore it defines a bundle $\vert f^{*}\mathscr{E} \vert \xrightarrow{\pi'} Y$ and this bundle is the pullback of $\pi$ and $f$. The composition $s \circ f$ yields a section as pictured below

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Now the condition $f^{*}(s)=0$ translates to the fact that the associated section we have constructed equals to the zero section of $\vert f^{*} \mathscr{E}\vert$. Let us denote the zero sections of both spaces by $0_S,0_Y$. If we define $V(s)$ as the pullback of $s,0_S$ we obtain the desired subspace and the factorization property is encoded in the universal property of the pullback. Furthermore this subscheme is unique.

This seems very nice theoretically but not very useful in practice. For the second part of the question I have to show that the Segre embedding of $\mathbb{P}^1_R \times_R \mathbb{P}^1_R \to \mathbb{P}^3_R$ arises as a vanishing as some $V(s)$.I know that it should be the vanishing locus of some homogeneous polynomial of degree 2 so I should be looking at $\mathcal{O}_{\mathbb{P}^3_R}(2)$ as a vector bundle. However with the tools that I have I don't see how I could possible translate the previous disgression into some actual computations because I don't know how to turn a global section of the sheaf into a geometrical section.

I know that morphisms into $ X \to \mathbb{P}^3_R$ are given by surjections $\mathcal{O}_X^4 \to \mathscr{L}$ with $\mathscr{L}$ an invertible sheaf. Can we link this description of the morphisms with the sections?

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