[[Abstract projective space]]
# Collineätion

A **collineätion** is an [[Morphism|isomorphism]]  $\phi:\mathcal{S}\to\mathcal{S}'$ of (abstract) [[Abstract projective space|projective spaces]] $\mathcal{S},\mathcal{S}'$, #m/def/geo 
i.e. a bijection which preserves the incidence of subspaces.
The set of all collineations is a [[group]], which for a [[projectivization]] is the [[Projective semilinear group]].

## Types of collineation

By the [[Fundamental theorem of projective geometry]], the only collineations of a projectivization are

- [[Projective semilinear group]]
  - [[Homography]] given by a linear transformation of the underlying vector space ([[Projective general linear group]])
  - [[Automorphic collineätion]] given by a coördinatewise [[Field automorphism]]

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