nLab morphism of projective spaces

Definition

Given a vector space $V$, denote by $p_V \colon V\backslash\{0\}\to P(V)$ the canonical projection onto the projective space of lines through origin.

Given two projective spaces $P(V), P(W)$ over a field (or a skewfield) $F$, a morphism $g \colon P(V)\to P(W)$ is a map defined on $P(V)\backslash P(f^{-1}(0))\to P(W)$ for some $F$-linear map $f: V\to W$ satisfying $g\circ p_V = p_W\circ f$ on $V\backslash f^{-1}(0)$.

If $f$ is an isomorphism then we say that $g$ is a projective isomorphism or homography.

Literature

• Marcel Berger, Géométrie, Cassini; Engl. translation: Geometry I, Springer 1987 (doi), section 4.5

category: geometry