Projective line - Knowlpedia

Definition

For a field $k$ , the projective line over $k$ is

\mathbb P_k^1=\operatorname{Proj}k[x_0,x_1].

Its $k$ -points are the one-dimensional subspaces of $k^2$ , written in homogeneous coordinates as $[x_0:x_1]$ , with $[x_0:x_1]=[\lambda x_0:\lambda x_1]$ for $\lambda\in k^\times$ .

Two affine charts

The open set $U_1=\{x_1\ne0\}$ has coordinate $z=x_0/x_1$ and is isomorphic to the affine line $\mathbb A_k^1$ . The other open set $U_0=\{x_0\ne0\}$ has coordinate $w=x_1/x_0$ . On their overlap,

w=z^{-1}.

Thus $\mathbb P_k^1$ is obtained by gluing two affine lines along their punctured affine lines by inversion. This choice of $z$ is the one used in the Riemann sphere and in the standard matrix formula for Möbius transformations.

The point at infinity

In the $z$ -chart, the complement is the single $k$ -point

\infty=[1:0],

so at the level of $k$ -points one often writes $\mathbb P^1(k)=k\cup\{\infty\}$ . The label $\infty$ depends on the chosen affine chart: no point of the projective line is intrinsically distinguished.

For $k=\mathbb C$ , the associated complex manifold is the Riemann sphere. Projective linear transformations act on $\mathbb P_k^1$ ; after coordinates are chosen, these are fractional linear transformations.

References

Robin Hartshorne, Algebraic Geometry, Springer, 1977. Publisher record. Relevant: Chapter I, §2 and Chapter II, §2.
Joe Harris, Algebraic Geometry: A First Course, Springer, 1992. Publisher record. Relevant: Lecture 1.