algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety projective toric variety; one-parameter subgroup; wall-crossing; semi-orthogonal decomposition; singularity categories; full exceptional collection; GIT; windows; geometric invariant theory; fan structure; bounded derived category of coherent sheaves; equivariant ample line bundle; VGIT; reductive linear algebraic group; variations of GIT structures; birational methods; blow-ups at smooth centres; linearisation of a group action 