Introduction to Geometric Invariant Theory

✍ Scribed by I.V. Dolgachev

• Publisher: Seoul National University
• Year: 1994
• Tongue: English
• Leaves: 147
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Dolgachev I.V. Introduction to geometric invariant theory (Seoul National University, 1994) ......Page 2
Notes of the Series of Lectures held at the Seoul National University ......Page 3
CONTENTS ......Page 5
Preface v ......Page 6
Introduction ix ......Page 7
Lecture 1. Algebraic groups 1 ......Page 10
Lecture 2. Algebraic group actions 9 ......Page 18
Lecture 3. Linearizations of actions 17 ......Page 26
Lecture 4. Quotients 30 ......Page 39
Lecture 5. Hilbert’s fourteenth problem 41 ......Page 50
Lecture 6. Stability 51 ......Page 60
Lecture 7. Numerical criterion of stability 60 ......Page 69
Lecture 8. Example: projective hypersurfaces 69 ......Page 78
Lecture 9. Example: configurations of linear subspaces 80 ......Page 89
Lecture 10. Toric varieties 97 ......Page 106
Lecture 11. Moduli space of curves 108 ......Page 117
References 136 ......Page 145
cover ......Page 1