# points on a surface and rank 2 vector bundles

The question title is a section of `Principles of algebraic geometry' by Griffiths and Harris, where it describes how rank 2 bundles can be reconstructed by looking at the vanishing points of sections on an algebraic surface. For integrable systems I would like to be able to carry out this construction and the associated residues in a totally explicit fashion (with transition functions etc.) for points on $$\mathbb{CP}^1\times \mathbb{CP}^1$$. The points would then encode momentum and the residues positions for solitons in a twistor-like construction (as $$\mathbb{CP}^1\times \mathbb{CP}^1$$ is a complexified projective Minkowski 2+1 space). My problem, with many apologies, is that I am not an expert in algebraic geometry, and while I can follow bits of the discussion in the book, I cannot figure out explicitly what is going on. I would be very grateful if someone could suggest somewhere where the construction is carried out in a explicit manner or provide other help on this. I would be happy to explain more about the integrable systems background if anyone wanted.

• This is called "Serre construction" in a lot of the literature; if you google that term, you get plenty of references with different degrees of expliciteness (although I can see that a lot of the refs only do the general homological algebra). Brinzanescu's "Holomorphic Vector Bundles over Compact Complex Surfaces" (Section 4.1) and Okonek-Schneider-Spindeler's "Vector bundles and complex projective spaces" (Chapter 1 Section 5) have longer sections on it which may have enough information. – Balazs Feb 23 at 17:47
• This article (and the references therein) might be useful: sciencedirect.com/science/article/pii/S1631073X11001257 In the case of $\mathbb{P}^1\times \mathbb{P}^1$ (i.e., $e=0$, with the notation of the paper), I presume that many things should simplify. – Pedro Montero Feb 23 at 18:06