Dolgachev I.V. Introduction to geometric invariant theory

Seoul National University, 1994. — 147 p. — (Lectures Notes, 25).These notes originate in a series of lectures given at the Tokyo Metropolitan University and Seoul National University in the Fall of 1993. These lectures have been extended into a graduate course at the University of Michigan in the Winter of 1994. Almost all of the material in these notes had been actually covered in my course. The main purpose of the notes is to provide a digest to Mumford’s book. Their sole novelty is the greater emphasis on dependence of the quotients on linearization of actions and also including toric varieties as examples of torus quotients of open subsets of affine space. We also briefly discuss Nagata’s counter-example to Hilbert’s Fourteenth Problem. Lack of time (and of interested audience) did not allow me to include such topic as the relationship between geometric invariant theory quotients and symplectic reductions. Only one application to moduli problem is included. This is Mumford’s construction of the moduli space of algebraic curves. The more knowledgeable reader will immediately recognize that the contents of these notes represent a small portion of material related to geometric invariant theory. Some compensation for this incompleteness can be found in a bibliography which directs the reader to additional results.
Only the last lecture assumes some advanced knowledge of algebraic geometry; the necessary background for all other lectures is the first two chapters of Shafarevich’s book. Because of arithmetical interests of some of my students, I did not want to assume that the ground field is algebraically closed, this led me to use more of the functorial approach to foundations of algebraic geometry.
I am grateful to everyone who attended my lectures in Tokyo, Seoul and Ann Arbor for their patience and critical remarks. I am especially thankful to Sarah-Marie Beicastro and Pierre Giguere for useful suggestions and corrections to preliminary version of these notes. I must also express great gratitude to Professor Uribe for organizing my visit to Tokyo Metropolitan University, and to my former students Jong Keum and Yonggu Kim for inviting me to Seoul National University and for their help in publishing these lecture notes.Algebraic groups
Algebraic group actions
Linearizations of actions
Quotients
Hilbert’s fourteenth problem
Stability
Numerical criterion of stability
Example: projective hypersurfaces
Example: configurations of linear subspaces
Toric varieties
Moduli space of curves