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Examples!
editThe end of the introductory section mentions the standard examples of affine and projective spaces -- it would be very useful to spell out what the fan/cones are for these explicitly! Including some diagrams, which should be simple enough in SVG, or can even be generated by sagemath.
I don't know these examples well enough myself yet to describe them with confidence, but may be able to help in a few months if no one else is able to add anything.