[[Collineätion]]
# Automorphic collineation

An **automorphic collineätion** is a [[Collineätion]] of a projective space given by a [[field automorphism]] applied coördinatewise, #m/def/geo
hence it is the action of $\Aut(\mathbb{K})$.

> [!check]- Proof of collineation
> It follows from
> $$
> \begin{align*}
> \left( \sum_{i=0}^n \lambda_{i} \mathbf{x}_{i} \right)^\sigma = \sum_{i=0}^n \lambda_{i}^\sigma \mathbf{x}_{i}^\sigma
> \end{align*}
> $$
> that $d$-dimensional linear subspaces are mapped to $d$-dimensional linear subspaces and containment/incidence is preserved.
> Hence $\sigma$ induces a collineation. 
> <span class="QED"/>
> 
## Properties

Consider the projective space $\mathrm{PG}(n, \mathbb{K})$.

1. [[Automorphic collineätion criterion]] (fixes basis elements)

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