Invariants of algebraic groups actions, an introduction.

Freie Universitaet, Berlin, 20 June - 15 July, 2011

Jaroslaw Wisniewski, Warsaw University

Summary: The course will concern algebraic group actions and their invariants. The first part of the course will be about polynomial invariants of linear actions of finite groups. Subsequently we will discuss algebraic actions of affine algebraic groups and their quotients.

Prerequisite: The course will be aimed at students who completed commutative algebra lecture (ring and fields) and an introductory course to algebraic geometry. Apart of taking part in lectures, the students are expected to solve homework problems and take part in exercise sessions.

Task: The course will serve as a slow pace introduction to lectures by Michel Brion and Nicolas Ressayre at the Autumn School in Algebraic Geometry Geometric Invariant Theory, old and new which is organized by Warsaw University with the support of Berlin Mathematical School in Lukecin, Poland, in September.

Schedule of classes: All lectures at Arnimalle 3, except SR 007/008 which is at Arnimalle 6.

First week: Wed, 22.06. 16:15-18:00, SR 119 and Thu, 23.06. 10:15-12:00, SR 210.
Second week: Tue, 28.06. 08:15-10:00, SR 119 and Wed, 29.06. 10:15-12:00, SR 005.
Third week: Tue, 05.07. 16:15-18:00, SR 007/008; Wed, 06.07. 16:15-18:00, SR 007/008, Thu, 07.07. 10:15-12:00, SR 210; Fri, 08.07, 14:15-16:00, SR 007/008.
Fourth week: Mon, 11.07. 10:15-12:00, SR 210; Tue 12.07. 16:15-18:00, SR 007/008; Wed 13.07, 16:15-18:00, SR 007/008; Thu, 14.07. 10:15-12:00, SR 210.

Topics

Linear actions of finite groups and their invariants: Gauss theorem about symmetric polynomials. Polynomial invariants, quasi invariants, homogenity. Reynolds operators. Finite generation of the ring of invariants. Linearly reductive groups. Counting the number of generators, Poincare series and Molien's theorem. Groups generated by pseudo-reflections, the theorem of Todd, Shepherd and Chevalley. Cohen-Macaulay graded algebras, rings of invariants are Cohen-Macaulay.
Actions of linear algebraic groups and their invariants: affine algebraic groups, linear groups, reductive groups, various characterisations. Chevalley theorem, homogeneous varieties. Geometric quotients, quotients of affine varieties, categorical quotients. Stability condition, quotient of the set of stable points is geometric.

Problem and exercise sheets:

First set: finite group actions, action of cyclic groups, symmetric polynomials, groups acting on the plane.
Second set: linearly reductive groups, theorem of Noether, counting the number of invariants.
Third set: Poincare-Hilbert series, Molien theorem, Cohen-Macaulay graded algebras.
Fourth set : linear groups, reductive groups, actions of C^* and C^+, their invariants and orbits.
Fifth, final set: five problems concerning groups actions and quotients; take-home exam.

Readings:

Cox, Little, O'Shea, Ideals, varieties, and algorithms. An introduction to computational algebraic geometry and commutative algebra. Third edition. Undergraduate Texts in Mathematics. Springer, New York, 2007.
Stanley, Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc. (N.S.) 1 (1979), online copy at Bull AMS
Benson, Polynomial invariants of finite groups. London Mathematical Society Lecture Note Series, 190. Cambridge University Press, Cambridge, 1993.
Sturmfels, Algorithms in invariant theory, Texts and Monographs in Symbolic Computations, Springer 1993.
Brion, Introduction to actions of algebraic groups, CIRM course, copy online at cedram
Dolgachev, Lectures on invariant theory. London Mathematical Society Lecture Note Series, 296. Cambridge University Press, Cambridge, 2003.
Mukai, An introduction to invariants and moduli, Cambridge Studies in Advanced Mathematics, 81. Cambridge University Press, Cambridge, 2003

Links:

Register at SAGE server at Warsaw University to work on examples.
Klaus Altmann's notes, BMS course Algebra.
Miles Reid page on surfaces including notes and problems about surface quotient singularities.