2003/08/07 by Igor Dolgachev · 9 citations
Computer Science · Mathematics · #Affine transformation #Algebra over a field #Algebraic Geometry and Number Theory #Algebraic number #Commutative Algebra and Its Applications #Function field of an algebraic variety #Geometric invariant theory #Invariant (physics) #Invariant theory #Mathematical analysis #Mathematics #Polynomial and algebraic computation #Pure mathematics #Quotient #Representation theory
paper · doi:10.1017/cbo9780511615436
openalex publication_date 2003/08/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
The primary goal of this 2003 book is to give a brief introduction to the main ideas of algebraic and geometric invariant theory. It assumes only a minimal background in algebraic geometry, algebra and representation theory. Topics covered include the symbolic method for computation of invariants on the space of homogeneous forms, the problem of finite-generatedness of the algebra of invariants, the theory of covariants and constructions of categorical and geometric quotients. Throughout, the emphasis is on concrete examples which originate in classical algebraic geometry. Based on lectures given at University of Michigan, Harvard University and Seoul National University, the book is written in an accessible style and contains many examples and exercises. A novel feature of the book is a discussion of possible linearizations of actions and the variation of quotients under the change of linearization. Also includes the construction of toric varieties as torus quotients of affine spaces.