Group action on a variety

Summary

An action of an algebraic group $G$ on a variety $X$ is a group action given by a morphism $G \times X \to X$. Orbits, stabilizers and fixed points of such actions are again algebraic objects. Quotients by such actions are the subject of geometric invariant theory. Many important varieties, such as projective spaces, Grassmannians and toric varieties, are best understood through a group action.