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I am trying to understand the theorem that characterizes morphisms to projective space as equivalent to the data of a line bundle together with global sections generating it.

I tried to find the corresponding line bundle associated with the Serge embedding, say from $\mathbb{P}^{1} \times \mathbb{P}^{1}$ to $\mathbb{P}^3$. But working directly with the definitions Hartshorne gives looks impractical. I don't know how to compute explicitly the global sections of the pullback of the twisting sheaf on $\mathbb{P}^3$. Is there any technique to compute the pullback of a line bundle, or more generally of any quasi coherent module in practice?

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The global sections of the twisting sheaf on $\mathbb{P}^3$ are just the coordinate functions on $\mathbb{P}^3$. The ambiguity in isomorphism of a fiber of the line bundle with $\mathbb{C}$ is what makes it a projective coordinate rather than an affine coordinate. So, if you know your embedding in terms of coordinates, you know the line bundle, or at least you know what its sections look like.

The Segre embedding is given by taking a point $([x_0,x_1],[y_0,y_1])$ to $[x_0y_0,x_0y_1,x_1y_0,x_1y_1]$. Therefore, the $x_0y_0$ is the restriction of a global section of the twisting sheaf to $\mathbb{P}^1 \times \mathbb{P}^1$ under the serge embedding, which, in terms of line bundles is a global section of $\mathcal{O}_{\mathbb{P}^1}(1) \otimes \mathcal{O}_{\mathbb{P}^1}(1)$ ( technically, the pullback of $\mathcal{O}_{\mathbb{P}^1}(1)$ under the two projections. )

Also, technically I've only described this in terms of global sections, but the whole description is natural over any open, so shows an isomorphism of sheaves (and therefore line bundles). I tried to describe it as one would think about it rather than all the fine details. Hope this helps.

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