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I'm studying momentum twistor variables, which I understand can be seen to be defined projectively from dual complexified Minkowski space to this complex projective space $\mathbb{C}\mathbb{P}^3$. When I try and search for resources to understand this space, I come up with a lot of rigorous looking maths that is slightly intimidating. I was hoping someone could provide a simple intuitive understanding of what this projective space is, and why it's useful for understanding momentum twistors.

I frequently see resources involving infinity twistors (Hodges, Mason, Skinner etc) but I don't understand what is so special about this space.

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