On Group Actions

Abstract

Publisher Summary A group action means an algebraic (regular) action of an algebraic group (or group scheme) on an algebraic variety (or scheme) that does not treat analytic actions of algebraic (or analytic) groups. The theory of torus (or toroidal) embeddings found many interesting applications in algebraic geometry. The significance of the theory lies in its ability of translate algebro-geometric problems involving torus embeddings into purely combinatorial problems. There is a similar trend of thinking in the theory of commutative algebras, as well. A torus is an algebraic k-group isomorphic to a product of several copies of the multiplicative group G m , and a unipotent group is an algebraic group that admits a k-monomorphism to the group of all upper-triangular unipotent matrices of some fixed size. The additive group is denoted by G a .