Field of Definition - Examples

Examples

  1. The zero-locus of x12+ x22 is both a Q-variety and a Qalg-algebraic set but neither a variety nor a Qalg-variety, since it is the union of the Qalg-varieties defined by the polynomials x1 + ix2 and x1 - ix2.
  2. With Fp(t) a transcendental extension of Fp, the polynomial x1p- t equals (x1 - t1/p) p in the polynomial ring (Fp(t))alg. The Fp(t)-algebraic set V defined by x1p- t is a variety; it is absolutely irreducible because it consists of a single point. But V is not defined over Fp(t), since V is also the zero-locus of x1 - t1/p.
  3. The complex projective line is a projective R-variety. (In fact, it is a variety with Q as its minimal field of definition.) Viewing the real projective line as being the equator on the Riemann sphere, the coordinate-wise action of complex conjugation on the complex projective line swaps points with the same longitude but opposite latitudes.
  4. The projective R-variety W defined by the homogeneous polynomial x12+ x22+ x32 is also a variety with minimal field of definition Q. The following map defines a C-isomorphism from the complex projective line to W: (a,b) → (2ab, a2-b2, -i(a2+b2)). Identifying W with the Riemann sphere using this map, the coordinate-wise action of complex conjugation on W interchanges opposite points of the sphere. The complex projective line cannot be R-isomorphic to W because the former has real points, points fixed by complex conjugation, while the latter does not.

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