Invariants groups actions: algebra and geometry

Seminar, 2012/2013

root system E8 source: Wikipedia

Summary: The seminar 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 seminar 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 the seminar, the students are expected to solve homework problems.

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.

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.
• Miles Reid page on surfaces including notes and problems about surface quotient singularities.