Examples Field of definition

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the complex projective line projective r-variety. (in fact, variety q minimal field of definition.) viewing real projective line being equator on riemann sphere, coordinate-wise action of complex conjugation on complex projective line swaps points same longitude opposite latitudes.
the projective r-variety w defined homogeneous polynomial x1+ x2+ x3 variety minimal field of definition q. following map defines c-isomorphism complex projective line w: (a,b) → (2ab, a-b, -i(a+b)). identifying w riemann sphere using map, coordinate-wise action of complex conjugation on w interchanges opposite points of sphere. complex projective line cannot r-isomorphic w because former has real points, points fixed complex conjugation, while latter not.






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