Linearization of a line bundle ($G\curvearrowright L,\pi (g\ell )=g\pi (\ell )$)

A linearization of a line bundle is a group action on its total space lifting the action on the base, linear on each fibre and obeying the group law. It induces actions on global sections of its tensor powers. For the projective space of lines in a linear representation $V$, the induced action on the dual of the tautological line bundle gives the natural linearization of $\mathcal{O}(1)$. Geometric invariant theory uses its invariant sections to define the semistable locus.