BIREP-Seminar: 

Kraft-Procesi reading course

This reading course will deal with the paper "Closures of conjugacy classes of matrices are normal" by Hanspeter Kraft and Claudio Procesi. The lectures are planned as follows:

- In addition there will be a talk on "Nori's desingularization and its singularities" by Roland Olbricht (Münster/Wuppertal/Grenoble) on Friday, 09.02.2007. 
Abstract: The quotient Z_n of the variety of n-dimensional modules of the free algebra in g generators is singular. M.V.Nori has in 1978 developed an approach to desingularise Z_n. This construction is connected to the variety of n^2-dimensional algebras. Unfortunately, it is smooth only if n=2. We discuss this case and determine the singular locus if n=3.

References

1. Hanspeter Kraft, Claudio Procesi: Closures of conjugacy classes of matrices are normal, Inventiones math. 53, 1979, pp. 227-247.
2. Hanspeter Kraft: Geometric methods in representation theory, Lecture Notes in Mathematics 944, Springer, 1982, pp. 180-258.
3. Hanspeter Kraft: Geometrische Methoden in der Invariantentheorie, 1984, Aspekte der Mathematik, Vieweg.
4. Claudio Procesi: The invariant theory of nxn matrices, Adv. Math. 19, 1976, pp. 306-381.


5. William Crawley-Boevey: Geometry of representations of algebras (pdf), Lecture Notes, Oxford University, 1993. (This is a survey of how algebraic geometry has been used to study representations of algebras (and quivers in particular)).
6. Joe Harris: Algebraic geometry: a first course, Graduate texts in mathematics 133, Springer, 1992.
7. Robin Hartshorne: Algebraic geometry, Springer, Graduate texts in mathematics 52, New York, 2000.
8. Hanspeter Kraft: Algebraic Transformation Groups. An Introduction (pdf). Lecture Notes, Universität Basel, February 2005. 
9. Ernst Kunz: Introduction to commutative algebra and algebraic geometry, Birkhäuser, 1985.
10. David Mumford: Geometric Invariant Theory, Erg. der Math. 34, Springer, 1970.
11. David Mumford: Introduction to Algebraic Geometry, Lecture notes.
    

Abstract

Plan

1. In the first talk basic definitions and the objects of study will be presented. We will speak on normal varieties in algebraic geometry and on Young diagrams, encoding nilpotent matrices.
2. Talks 2 & 3 are the core of the course, presenting the main results and the following method: 
   It consists of constructing an auxiliary variety Z which is a complete intersection and of which the closure of C_A is a quotient. Then Z, and hence its quotient, will be shown to be normal.
3. Talks 4 to 7 will provide lemmas to prove the main theorem presented in talk 2. One idea is the use of a special case of the theory of vector space crowns, namely nilpotent pairs and the representations of their orbits. The variety Z will be stratified and one gets a formula for the dimension of each strata. This will then help to satisfy the conditions of Serre's criterion to get the result concerning normality of Z.

Prerequisites

Definitions / Results

• A variety X is called irreducible if for any decomposition X=X_1 U X_2 of X into a union of two subvarities, either X_1=X or X_2=X. In other words, there are no non-trivial decompositions of X into smaller varieties.
• If X is irreducible, then the coordinate ring of X, O(X), is an integral domain.
• An affine variety X is normal if X is irreducible and O(X) is integrally closed (i.e. if x in the field of fractions of O(X) is integral over O(X), then x lies in O(X)).
• X is normal in a point x, if the local ring O_X,x is integrally closed. (O_X,x is the localization of O(X) at the maximal ideal m.)
• The set of nilpotent matrices, the nilpotent cone in M_n, is a normal variety.
• Algebraic quotients (see chapter 5, p. 234-235 of reference 2. (Kraft)).
• A quotient of a normal variety is also normal.
• A special case of pre-images are the fibers of a morphism f: X --> Y. Let y be in Y. Then f^-1(y):={x in X : f(x)=y} is called the fiber of y in Y. The fiber of a point y in Y can be empty. Note: f( g A g^−1) = g f(A) g^−1 for g in GL_n.
• The pull-back of a regular function is again regular.