2.2 The First Examples

A complex projective line is homeomorphic to the two-dimensional sphere.⁴ The set of real points of a real projective line is homeomorphic to a circle; by the Jordan theorem it divides the complexification. Therefore a real projective line is of type I. It has a pair of complex orientations, but they do not add anything, since the real line is connected and admits only one pair of orientations opposite to each other.

The action of $conj$ on the set of complex points of a real projective line is determined from this picture by rough topological arguments. Indeed, it is not difficult to prove that any smooth involution of a two-dimensional sphere with one-dimensional (and non-empty) fixed point set is conjugate in the group of autohomeomorphisms of the sphere to the symmetry in a plane. - $(^{\arabic{endnotectr}})$endnotei Proof of this purely topological theorem: The orbit space of such an involution $\tau :S^2\to S^2$ has to be a compact connected surface with non-empty boundary and Euler characteristic $1=\frac12\chi(S^2)$. By topological classification of compact surfaces, any surface with these properties is homeomorphic to $D^2$. Therefore the fixed point set is homeomorphic to circle. By Jordan-Schoenflies Theorem, it divides the sphere into two discs. The discs are mapped to each other by the involution, since the complement of the fixed point set covers the corresponding part (the interior) of the orbit space and any two-fold covering of a disc is trivial. A homeomorphism conjugating $\tau$ with the symmetry $\tau :S^2\to S^2: (x_1,x_2,x_3)\mapsto(-x_1,x_2,x_3)$ is constructed now as follows: take any homeomorphism $h$ of the disc bounded by the fixed point set to the half-sphere $\{x\in S^2 : x_3\ge0\}$. Extend it to the complementary disc by the formula $h(x)=\sigma h\tau (x)$. It is clear that $\tau =h^{-1}\sigma h$.

The set of complex points of a nonsingular plane projective conic is homeomorphic to $S^2$, because the stereographic projection from any point of a conic to a projective line is a homeomorphism. Certainly, an empty conic, as any real algebraic curve with empty set of real points, is of type II. The empty set cannot divide the set of complex points. For the same reasons as a line (i.e. by Jordan theorem), a real nonsingular curve of degree 2 with non-empty set of real points is of type I. Thus the real scheme $\langle 1\rangle$ of degree 2 is of type I, while the scheme $\langle 0\rangle$ is of type II for any degree.